Mathematical Constants
compiled by Stanislav Sýkora, Extra Byte, Via R.Sanzio 22C, Castano Primo, Italy 20022
in Stan's Library, Ed.S.Sykora, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001
This page is dedicated to my late teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. 
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This is a constant-at-a-glance list. It keeps growing, so keep coming back.
Bold dots after a value are links to the OEIS database.
Basic math constants 
Zero and One (and i, and ...)  0 and 1 (and √(-1), and ...)  Can anything be more basic than these two ? (or three, or ...) 
π, Archimedes' constant  3.141 592 653 589 793 238 462 643 •••  Circumference of a disc with unit diameter. 
e, Euler number, Napier's constant  2.718 281 828 459 045 235 360 287 •••  Base of natural logarithms. 
γ, Euler-Mascheroni constant  0.577 215 664 901 532 860 606 512 •••  Limit[n→∞]{(1+1/2+1/3+...1/n) - ln(n)} 
√2, Pythagora's constant  1.414 213 562 373 095 048 801 688 •••  Diagonal of a square with unit side. 
Φ, Golden ratio  1.618 033 988 749 894 848 204 586 •••  Φ = (√5 + 1)/2 = 2.cos(π/5). Diagonal of a unit side pentagon. 
φ, Inverse golden ratio (often confused with Φ)  0.618 033 988 749 894 848 204 586 •••  φ = 1/Φ = Φ -1 =(1-φ)/φ or φ = (√5 - 1)/2. 
δs, Silver ratio / mean  2.414 213 562 373 095 048 801 688 •••  δs = 1+√2. 
Constants derived from the basic ones 
Conversions between logarithms for bases 2, e, 10: 
ln(2), Natural logarithm of 2  0.693 147 180 559 945 309 417 232 •••  ex = 2
log(2), Decadic logarithm of 2  0.301 029 995 663 981 195 213 738 •••  10x = 2
ln(10), Natural logarithm of 10  2.302 585 092 994 045 684 017 991 •••  ex = 10
ln2(10), Binary logarithm of 10  3.321 928 094 887 362 347 870 319 •••  2x = 10
log(e), Decadic logarithm of e  0.434 294 481 903 251 827 651 128 •••  10x = e
ln2(e), Binary logarithm of e  1.442 695 040 888 963 407 359 924 •••  2x = e
Spin-offs of the imaginary unit i. Formally, i is a solution of z2 = -1 and of z = e zπ/2. For any integer k and any z, i 4k+z = i z. i 4f = e i2πf 
De Moivre numbers ei2πk/n  cos(2πk/n) + i.sin(2πk/n)  for any integer k and n≠0. 
ii = e-π/2 , the imaginary unit elevated to itself  0.207 879 576 350 761 908 546 955 •••  A transcendental real number 
i-i = (-1)-i/2 = eπ/2  4.810 477 380 965 351 655 473 035 •••  Inverse of the above. Square root of Gelfond's constant. 
ln(i) / i = π/2  1.570 796 326 794 896 619 231 321 •••  This value could also be classified as a π spin-off 
i! = Γ(1+i) = i*Γ(i) (see Gamma function)  0.498 015 668 118 356 042 713 691 •••  -i 0.154 949 828 301 810 685 124 955 ••• 
| i! | absolute value of the above  0.521 564 046 864 939 841 158 180 •••  arg( i! ) = - 0.301 640 320 467 533 197 887 531 ••• rad 
i^i^i^... infinite power tower of i; solution of z = i z  0.438 282 936 727 032 111 626 975 •••  +i 0.360 592 471 871 385 485 952 940 ••• 
| i^i^i | absolute value of the above  0.567 555 163 306 957 825 384 613 •••  arg( i^i^i^... ) = 0.688 453 227 107 702 130 498 767 ••• rad 
Basic roots of i, up to a term of 4k in the exponent (like i4k+1/4 = i1/4, for any integer k):
i1/2 = √i = (1 + i)/√2 = cos(π/4) + i.sin(π/4)  0.707 106 781 186 547 524 400 844 •••  +i 0.707 106 781 186 547 524 400 844 ••• 
i1/3 = (√3 + i)/2 = cos(π/6) + i.sin(π/6)  0.866 025 403 784 438 646 763 723 •••  +i 0.5 
i1/4 = cos(π/8) + i.sin(π/8)  0.923 879 532 511 286 756 128 183 •••  +i 0.382 683 432 365 089 771 728 459 ••• 
i1/5 = cos(π/10) + i.sin(π/10)  0.951 056 516 295 153 572 116 439 •••  +i 0.309 016 994 374 947 424 102 293 ••• 
i1/6 = cos(π/12) + i.sin(π/12)  0.965 925 826 289 068 2867 497 431 •••  +i 0.258 819 045 102 520 762 348 898 ••• 
i1/7 = cos(π/14) + i.sin(π/14)  0.974 927 912 181 823 607 018 131 ...  +i 0.222 520 933 956 314 404 288 902 ... 
i1/8 = cos(π/16) + i.sin(π/16)  0.980 785 280 403 230 449 126 182 ...  +i 0.195 090 322 016 128 267 848 284 ... 
i1/9 = cos(π/18) + i.sin(π/18)  0.984 807 753 012 208 059 366 743 ...  +i 0.173 648 177 666 930 348 851 716 ••• 
i1/10 = cos(π/20) + i.sin(π/20)  0.987 688 340 595 137 726 190 040 •••  +i 0.156 434 465 040 230 869 010 105 ••• 
e spin-offs; note also: e = (e1/e)^(e1/e)^(e1/e)^... = PowerTower(e1/e) 
2e  5.436 563 656 918 090 470 720 574 •••  1/e = 0.367 879 441 171 442 321 595 523 ••• 
cosh(1) = (e + 1/e)/2  1.543 080 634 815 243 778 477 905 •••  sinh(1) = (e - 1/e)/2 = 1.175 201 193 643 801 456 882 381 ••• 
e2  7.389 056 098 930 650 227 230 427 •••  1/e2 = 0.135 335 283 236 612 691 893 999 ••• 
√e  1.648 721 270 700 128 146 848 650 •••  1/√e = 0.606 530 659 712 633 423 603 799 ••• 
e±i = cos(1) ± i sin(1) = cosh(i) ± sinh(i)  0.540 302 305 868 139 717 400 936 •••  ±i 0.841 470 984 807 896 506 652 502 ••• 
ee 15.154 262 241 479 264 189 760 430 •••  e-e = 0.065 988 035 845 312 537 0767 901 ••• 
e±ie = cos(e) ± i.sin(e)  - 0.911 733 914 786 965 097 893 717 •••  ±i 0.410 781 290 502 908 695 476 009 ••• 
ie = cos(eπ/2) ± i.sin(eπ/2)  -0.428 219 773 413 827 753 760 262 •••  ±i -0.903 674 623 776 395 536 600 853 ••• 
e1/e 1.444 667 861 009 766 133 658 339 •••  e-1/e = 0.692 200 627 555 346 353 865 421 ••• 
e±i/e = cos(1/e) ± i.sin(1/e)  0.933 092 075 598 208 563 540 410 •••  ±i 0.359 637 565 412 495 577 0382 503 ••• 
Infinite power tower of 1/e  0.567 143 290 409 783 872 999 968 ...  (1/e)^(1/e)^(1/e)^...; also solution of x = e-x 
Ramanujan's number:     262537412640768743 +  0.999 999 999 999 250 072 597 198 •••  exp(π√163). Closest approach of exp(π√n) to integer. 
π spin-offs 
2π  6.283 185 307 179 586 476 925 286 •••  1/π = 0.318 309 886 183 790 671 537 767 ••• 
π2  9.869 604 401 089 358 618 834 490 •••  1/π2 = 0.101 321 183 642 337 771 443 879 ••• 
√π  1.772 453 850 905 516 027 298 167 •••  1/√π = 0.564 189 583 547 756 286 948 079 ••• 
ln(π)  1.144 729 885 849 400 174 143 427 •••  log10(π) = 0.497 149 872 694 133 854 351 268 ••• 
ln(π).π  3.596 274 999 729 158 198 086 001 ...  ln(π)/π = 0.364 378 839 675 906 257 049 587 ... 
π±i = cos(ln(π)) ± i.sin(ln(π))  0.413 292 116 101 594 336 626 628 ...  ±i 0.910 598 499 212 614 707 060 044 ... 
iπ = cos(π2/2) + i.sin(π2/2)  0.220 584 040 749 698 088 668 945 ...  - i 0.975 367 972 083 631 385 157 482 ... 
ππ 36.462 159 607 207 911 770 990 826 •••  π = 0.027 425 693 123 298 106 119 556 ••• 
π±iπ = cos(π.ln(π)) ± i.sin(π.ln(π))  -0.898 400 579 757 743 645 668 580 ...  ±i -0.439 176 955 555 445 894 369 454 ... 
π1/π 1.439 619 495 847 590 688 336 490 •••  π-1/π = 0.694 627 992 246 826 153 124 383 ••• 
π±i/π = cos(ln(π)/π) ± i.sin(ln(π)/π)  0.934 345 303 678 637 694 262 240 ...  ±i 0.356 368 985 033 313 899 907 691 ... 
e and π combinations, except trivial ones like, for any integer k, eiπk = (-1)k, cosh(iπk) = (-1)k, sinh(iπk) = 0 
  8.539 734 222 673 567 065 463 550 •••  √() = 2.922 282 365 322 277 864 541 623 ••• 
e/π  0.865 255 979 432 265 087 217 774 •••  π/e = 1.155 727 349 790 921 717 910 093 ••• 
eπ = (-1)-i, Gelfond's constant 23.140 692 632 779 269 005 729 086 •••  e = 0.043 213 918 263 772 249 774 417 ••• 
e1/π 1.374 802 227 439 358 631 782 821 •••  e-1/π = 0.727 377 349 295 216 469 724 148 ... 
e±i/π = cos(1/π) ± i.sin(1/π) 0.949 765 715 381 638 659 994 406 ...  ±i 0.312 961 796 207 786 590 745 276 ... 
πe 22.459 157 718 361 045 473 427 152 •••  π-e = 0.044 525 267 266 922 906 151 352 ••• 
π1/e 1.523 671 054 858 931 718 386 285 •••  π-1/e = 0.656 309 639 020 204 707 493 834 ... 
γ spin-offs and some combinations (for basic definition of γ, see the Basic Constants section) 
  1.154 431 329 803 065 721 213 024 ...  1/γ = 1.732 454 714 600 633 473 583 025 ••• 
ln(γ)  -0.549 539 312 981 644 822 337 661 •••  log10(γ) = -2.386 618 912 168 323 894 602 884 ... 
eγ  1.569 034 853 003 742 285 079 907 ...  πγ = 1.813 376 492 391 603 499 613 134 ••• 
eγ  1.781 072 417 990 197 985 236 504 •••  e = 0.561 459 483 566 885 169 824 143 ••• 
e±iγ = cos(γ) ± i sin(γ)  0.837 985 287 880 196 539 954 992 •••  ±i 0.545 692 823 203 992 788 157 356 ••• 
Infinite power tower of γ  0.685 947 035 167 428 481 875 735 ...  γ^γ^γ^...; also solution of x = γx 
Golden ratio spin-offs and combinations (for basic definition of Φ and its inverse φ, see the Basic Constants section) 
Complex golden ratio Φc = 2.eiπ/5  1.618 033 988 749 894 848 204 586 •••  +i 1.175 570 504 584 946 258 337 411 ••• 
Associate of Φ = imaginary part of Φc  1.175 570 504 584 946 258 337 411 •••  2.sin(π/5), while Φ = 2.cos(π/5) = real part of Φc 
Square root of Φ  1.272 019 649 514 068 964 252 422 •••  √Φ; relates the sides of squares with golden-ratio areas. 
Square root of the inverse φ  0.786 151 377 757 423 286 069 559 •••  1/√Φ 
Cubic root of Φ  1.173 984 996 705 328 509 966 683 •••  √Φ1/3. Relates edges of cubes with golden-ratio volumes.
Cubic root of the inverse φ  0.851 799 642 079 242 917 055 213 ...  1/√Φ1/3
π/Φ = π.φ  1.941 611 038 725 466 577 346 865 •••  Area of golden ellipse with semi_axes {1,φ}
ln(Φ) = - ln(φ)  0.481 211 825 059 603 447 497 758 •••  Natural logarithm of Φ 
Φ 2/π, such as in golden spiral  1.358 456 274 182 988 435 206 180 •••  (2/π) ln(Φ) = 0.306 348 962 530 033 122 115 675 ••• 
Infinite power tower of the inverse φ  0.710 439 287 156 503 188 669 345 ...  φ^φ^φ^...; also solution of x = φx = Φ-x 
Miscellaneous derived constants: 
2^√2, the Gelfond - Schneider constant  2.665 144 142 690 225 188 650 297 •••  a transcendental number ... 
√2^√2 = 2^(1/√2)  1.632 526 919 438 152 844 773 495 •••  ... and its root, also transcendental 
Classical, named math constants 
Apéry's constant ζ(3)  1.202 056 903 159 594 285 399 738 •••  A special value of the Riemann function ζ(x) 
Artin's constant  0.373 955 813 619 202 288 054 728 •••  Product of factors [1-1/p(p-1)], p prime 
Bernstein's constant β  0.280 169 499 023 869 133 036 436 •••  From the theory of function approximations by polynomials 
Blazy's constant  2.566 543 832 171 388 844 467 529 ...  its Don Blazy's expansion generates all prime numbers 
Brun's constant for twin primes B4  1.902 160 583 104 ••• (?)  Sum of reciprocals of prime pairs (p,p+2) 
Brun's constant for prime cousins B4  1.197 0449 ••• (?)  Sum of reciprocals of prime pairs (p,p+4) 
Brun's constant for prime quadruplets B'4  0.870 588 380 ... (?)  Sum of reciprocals of prime quadruplets (p,p+2,p+4,p+6) 
Champernowne constant C10  0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...  String concatenation of dec expansions of natural numbers 
Catalan's constant C  0.915 965 594 177 219 015 054 603 •••  C = Sum[n=0,∞]{(-1)^n/(2n+1)^2} 
Continued fractions constant  1.030 640 834 100 712 935 881 776 •••  (1/6)π2/(ln(2)ln(10)). Mean c.f.terms per decimal digit 
Conway's constant λ(3)  1.303 577 269 034 296 391 257 099 •••  Growth rate of derived look-and-say strings 
Delian's constant  1.259 921 049 894 873 164 767 210 •••  21/3 
Embree - Trefethen constant β  0.70258 ••• (?)  Theory of 2nd order recurrences with random add/subtract 
Erdös - Borwein constant  1.606 695 152 415 291 763 783 301 •••  Sum[n=1,∞]{1/(2^n -1)} 
Feigenbaum reduction parameter α  -2.502 907 875 095 892 822 283 902 •••  Appears in the theory of chaos 
Feigenbaum bifurcation velocity δ  4.669 201 609 102 990 671 853 203 •••  Appears in the theory of chaos 
Gauss' constant G  0.834 626 841 674 073 186 814 297 •••  1/agm(1,√2); agm ... arithmetic-geometric mean 
Gauss-Kuzmin-Wirsing constant λ1  0.303 663 002 898 732 658 597 448 •••  2nd eigenvalue of GKW functional operator (first is 1) 
Gelfond's constant  23.140 692 632 779 269 005 729 086 •••  e^π = (-1)^(-i) 
Gelfond - Schneider constant  2.665 144 142 690 225 188 650 297 •••  2^√2, a transcendental number 
Gibbs constant G  1.178 979 744 472 167 270 232 028 •••  (2/π)(Wilbraham-Gibbs constant G'); see below. 
Glaisher-Kinkelin constant A  1.282 427 129 100 622 636 875 342 •••  Appears in number theory 
Golomb-Dickman constant λ  0.624 329 988 543 550 870 992 936 •••  Longest cycle distribution in random permutations 
Gompertz constant G  0.596 347 362 323 194 074 341 078 •••  -e.Ei(-1), Ei(x) being the exponential integral 
Grossmann's constant  0.737 338 303 369 29 ••• (?)  The only x for which {a0=1; a1=x; an+2=an/(1+an+1)} converges 
Khinchin-Lévy constant γ  3.275 822 918 721 811 159 787 681 •••  exp(π2/(12.ln2)); unstable nomenclature 
Lévy constant γ  1.186 569 110 415 625 452 821 722 •••  π2/(12.ln2). Logarithm of Khinchin-Lévy's 
Knuth's random-generators constant  0.211 324 865 405 187 117 745 425 ...  (1-sqrt(1/3))/2; also solid angle of a cone with magic polar angle 
Kolakoski constant γ  0.794 507 192 779 479 276 240 362 •••  Related to Kolakoski sequence 
Lemniscate constant L  2.622 057 554 292 119 810 464 839 •••  L = πG, where G is the Gauss' constant 
First lemniscate constant L1  1.311 028 777 146 059 905 232 419 •••  L1 = L/2 = πG/2 
Second lemniscate constant L2  0.599 070 117 367 796 103 337 484 •••  L2 = 1/(2G) 
Landau-Ramanujan constant  0.764 223 653 589 220 662 990 698 •••  Related to the density of sums of two integer squares
Laplace limit constant λ  0.662 743 419 349 181 580 974 742 •••  Let η = √(1+λ2); then λeη = 1+η Click here for more
Liouville's constant  0.110 001 000 000 000 000 000 001 •••  Sum[n=0,∞]{10^(-n!)} 
Madelung's constant M3  -1.747 564 594 633 182 190 636 212 •••  M3 = Sum[i,j,k]{(-1)^(i+j+k)/sqr(i^2+j^2+k^2)} 
Meissel-Merten's constant B1  0.261 497 212 847 642 783 755 426 •••  Limit[n→∞]{Sum[prime p≤n]{1/p}-ln(ln(n))} 
MRB constant  0.187 859 642 462 067 120 248 517 •••  Sum[n=1,2,...]{(-1)^n (n^(1/n) - 1)} 
MKB constant  0.687 652 368 927 694 369 809 3 •••  Limit[n→∞]{abs(Intg[1,2n]{(-1)^x x^(1/x) dx]}} 
Omega constant W(1)  0.567 143 290 409 783 872 999 968 •••  Root of [x - e-x] or [x + ln(x)]. See also. 
Otter's constant α  2.955 765 285 651 994 974 714 817 •••  Appears in enumeration of rooted and unrooted trees: 
Otter's asymptotic constant βu for unrooted trees  0.534 949 606 1(?) •••  UT(n) ~ βu αn n-5/2
Otter's asymptotic constant βr for rooted trees  0.439 924 012 571 (?) •••  RT(n) ~ βr αn n-3/2 (V. Kotesovec)
Plastic number ρ (or silver constant)  1.324 717 957 244 746 025 960 908 •••  Real root of x3 = x + 1 
Pogson's ratio  2.511 886 431 509 580 111 085 032 •••  1001/5; in astronomy 1 stellar magnitude brightness ratio 
Polya's random-walk constant p3  0.340 537 329 550 999 142 826 273 •••  Probability a 3D-lattice random walk returns back. See also 
Prévost's constant  3.359 885 666 243 177 553 172 011 •••  Sum of reciprocals of Fibonacci numbers 
Reciprocal even Fibonacci constant  1.535 370 508 836 252 985 029 852 •••  Sum of reciprocals of even-indexed Fibonacci numbers 
Reciprocal odd Fibonacci constant  1.824 515 157 406 924 568 142 158 •••  Sum of reciprocals of odd-indexed Fibonacci numbers 
Rényi's parking constant m  0.747 597 920 253 411 435 178 730 •••  Density of randomly parked cars in a street 
Sierpinski's constant K  2.584 981 759 579 253 217 065 893 •••  For explanation, click also here 
Soldner's constant μ  1.451 369 234 883 381 050 283 968 •••  Root of logarithmic integral li(x) 
Somos' quadratic recurrence constant  1.661 687 949 633 594 121 295 818 •••  √(1√(2√(3√(4 ...)))) 
Shall-Wilson or twin primes constant C2  0.660 161 815 846 869 573 927 812 •••  Product[twin_primes p,p+2]{p(p-2)/(p-1)2} 
Theodorus' constant  1.732 050 807 568 877 293 527 446 •••  √3. 
Viswanath's constant  1.131 988 248 794 3 ••• (?)  Growth of Fibonacci-like sequences with random add/subtract 
Wilbraham-Gibbs constant G'  1.851 937 051 982 466 170 361 053 •••  Intg[0,π]{sin(θ)/θ dθ}. 
Some other, notable math constants 
(1-1/2)*(1-1/4)*(1-1/8)*(1-1/16)* ...  0.288 788 095 086 602 421 278 899 •••  Product [k=1,∞](1-xk), for x=1/2 
1+1/22+1/33+1/44+ ...  1.291 285 997 062 663 540 407 282 •••  Sum [k=1,∞](1/kk) 
Infinite power towers for inverses of some sequences a(n): iPT{a(n)} = Lim[n→∞]{(1/a(k)^(1/a(k+1)^(1/a(k+2)^...^(1/a(n))}, k being the first index such that a(k)>1 
iPT{2}, the infinite power tower of 1/2  0.641 185 744 504 985 984 486 200 ...  (1/2)^(1/2)^(1^2)^...; also solution of x = 2-x 
iPT{n}  0.690 347 126 114 964 319 467 328 ...  (1/2)^(1/3)^(1/4)^(1/5)^... 
iPT{2^n}  0.570 203 397 398 373 262 211 917 ...  (1/2)^(1/4)^(1/8)^(1/16)^... 
iPT{prime(n)}  0.719 405 031 245 092 118 809 081 ...  (1/2)^(1/3)^(1/5)^(1/7)^(1/11)^... 
iPT{n!}  0.885 946 913 356 773 110 697 240 ...  (1/2!)^(1/3!)^(1/4!)^(1/5!)^(1/6!)^... 
Selected natural and integer numbers and their sequences 
Large integers 
Googol  10100 = 10^100 A large integer ... 
Googolplex  10googol = 10^10^100 ... a larger integer ... 
Googolplexplex  10googolplex = 10^10^10^100 ... and a still larger one. 
Graham's number (last 20 digits)  ... 04575627262464195387  3^^^...^^^3, with 64 power operators; too big to write 
Skewes' numbers  10^14 < n < e^e^e^79  Bounds on the first integer for which π(n) < li(n) 
Polygonal (2D) figurate numbers. See also on OEIS: numbers which are (A090466) or aren't (A090467) "some" polygonal number. See the link for more! 
Triangualar numbers  1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, •••  n(n+1)/2 
Square numbers  1, 4, 9, 16, 25, 36, 49, 64, 81,100,121, •••  n*n 
Square triangular numbers  1,36,1225,41616,1413721,48024900, •••  [[(3+2√2)k-(3-2√2)k]/(4√2)]2; both triangular and square 
Pentagonal numbers  1, 5, 12, 22, 35, 51, 70, 92, 117, 145, •••  n(3n-1)/2 
Hexagonal numbers  1, 6, 15, 28, 45, 66, 91, 120, 153, 190, •••  n(2n-1); also cornered hexagonal numbers 
Heptagonal numbers  1, 7, 18, 34, 55, 81, 112, 148, 189, •••  n(5n-3)/2 
Octagonal numbers  1, 8, 21, 40, 65, 96, 133, 176, 225, •••  n(3n-2) 
Centered polygonal (2D) figurate numbers 
Centered triangular numbers  1, 4, 10, 19, 31, 46, 64, 85, 109, 136, •••  (3n2+3n+2)/2. See also: centered triangular primes ••• 
Centered square numbers  1, 5, 13, 25, 41, 61, 85, 113, 145, 181, •••  2n2-2n+1. See also: centered square primes ••• 
Centered pentagonal numbers  1, 6, 16, 31, 51, 76, 106, 141, 181, 226, •••  (5(n-1)2+5(n-1)+2)/2. See also: centered pentagonal primes ••• 
Centered hexagonal numbers  1, 7, 19, 37, 61, 91, 127, 169, 217, •••  n3- (n-1)3 = 3n(n-1)+1; also hex numbers 
Centered heptagonal numbers  1, 8, 22, 43, 71, 106, 148, 197, 253, •••  (7n2-7n+2)/2 
Centered octagonal numbers  1, 9, 25, 49, 81, 121, 169, 225, 289, •••  (2n-1)2; squares of odd numbers 
Polyhedral (3D) figurate numbers 
Tetrahedral numbers Tn  1, 4, 10, 20, 35, 56, 84, 120, 165, 220, •••  n(n+1)(n+2)/6. Only three are squares: 1, 4, 19600 
Cubic numbers, cubes  1, 8, 27, 64, 125, 216, 343, 512, 729, •••  n3 
Pentatopic (or pentachoron) numbers  0, 1, 5, 15, 35, 70, 126, 210, 330, 495, •••  n(n+1)(n+2)(n+3)/24; n=0,1,2,... 
Octahedral numbers On  1, 6, 19, 44, 85, 146, 231, 344, 489, •••  n(2n2+1)/3. 
Combinatorial numbers and basic counting 
Binomial coefficients C(n,m) = n!/(m!(n-m)!) (ways to pick m among n labelled elements); C(n,m)=0 if m<0 or m>n; C(n,0)=1; C(n,1)=n; C(n,m)=C(n,n-m): 
m = 2, n = 4,5,6,...   6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, •••  n(n-1)/2; shifted triangular numbers 
m = 3, n = 6,7,8,...   20, 35, 56, 84, 120, 165, 220, 286, 364, •••  n(n-1)(n-2)/3!; shifted tetrahedral numbers 
m = 4, n = 8,9,10,...   70, 126, 210, 330, 495, 715, 1001, 1365, •••  n(n-1)(n-2)(n-3)/4!; for n < 2m, use C(n,n-m) 
m = 5, n = 10,11,12...   252, 462, 792, 1287, 2002, 3003, 4368, •••  n(n-1)(n-2)(n-3)(n-4)/5! 
m = 6, n = 12,13,14,...   924, 1716, 3003, 5005, 8008, 12376, •••  n(n-1)(n-2)(n-3)(n-4)(n-5)/6! = n(6)/6! 
m = 7, n = 14,15,16...   3432, 6435, 11440, 19448, 31824, •••  n(7)/7!; all cases up to n=14 are covered 
Central binomial coefficients C(2n,n) = (2n)!/n!2  1, 2, 6, 20, 70, 252, 924, 3432, 12870, •••  C(2n,n) = Sum[k=0,n]{C2(n,k)}: Franel number of order 2 
Factorials n! = 1*2*3...*n  1, 1, 2, 6, 24, 120, 720, 5040, 40320, •••  Permutations of an ordered set of n labelled elements 
   Quadruple factorials (2n)!/n!  1, 2, 12, 120, 1680, 30240, 665280, •••  in terms of Catalan numbers, equal to (n+1)!C(n) 
Factorions in base 10  1, 2, 145, 40585 (that's all)  Equal to the sum of factorials of their dec digits 
Factorions in base 16  1, 2, 2615428934649 (that's all)  Equal to the sum of factorials of their hex digits 
Franel numbers of order 3  1, 2, 10, 56, 346, 2252, 15184, 104960, •••  Sum[k=0,n]{C3(n,k)} 
Lah numbers L(n,m) (unsigned); signed L(n,m) = (-1)nL(n,m); They expand rising factorials in terms of falling factorials and vice versa. L(n,1) = n!  
m = 2, n = 2,3,4,...   1, 6, 36, 240, 1800, 15120, 141120, •••   
m = 3, n = 3,4,5,...   1, 12, 120, 1200, 12600, 141120, •••  General formula: L(n,m)=C(n,m)(n-1)!/(m-1)! 
m = 4, n = 4,5,6,...   1, 20, 300, 4200, 58800, 846720, •••   
Stirling numbers of the first kind c(n,m) (unsigned); signed s(n,m) = (-1)n-mc(n,m); number of permutations of n distinct elements with exactly m cycles.  
m = 1, n = 1,2,3,...   1, 1, 2, 6, 24, 120, 720, 5040, 40320, •••  (n-1)! Note: by convention, c(n,0) = 1 
m = 2, n = 2,3,4,...   1, 3, 11, 50, 274, 1764, 13068, 109584, •••  a(n+1)=n*a(n)+(n-1)! 
m = 3, n = 3,4,5,...   1, 6, 35, 225, 1624, 13132, 118124, •••   
m = 4, n = 4,5,6,...   1, 10, 85, 735, 6769, 67284, 723680, •••  A definition of s(n,m): 
m = 5, n = 5,6,7,...   1, 15, 175, 1960, 22449, 269325, •••  x(n) = x(x-1)(x-2)...(x-(n-1)) = Sum{m=0:n}[s(n,m).xm] 
m = 6, n = 6,7,8,...   1, 21, 322, 4536, 63273, 902055, •••  See also OEIS A008275 
m = 7, n = 7,8,9,...   1, 28, 546, 9450, 157773, 2637558, •••   
m = 8, n = 8,9,10,...   1, 36, 870, 18150, 357423, 6926634, ...   
m = 9, n = 9,10,11,...   1, 45, 1320, 32670, 749463, 16669653, ...   
Stirling numbers of the second kind S(n,m); number of partitions of n distinct elements into m non-empty subsets. S(n,1) = 1. By convention, S(0,0) = 1.  
m = 2, n = 2,3,4,...   1, 3, 7, 15, 31, 63, 127,255, 511, 1023, •••  2(n-1)-1 
m = 3, n = 3,4,5,...   1, 6, 25, 90, 301, 966, 3025, 9330, •••   
m = 4, n = 4,5,6,...   1, 10, 65, 350, 1701, 7770, 34105, •••  A definition of S(n,m): 
m = 5, n = 5,6,7,...   1, 15, 140, 1050, 6951, 42525, 246730, •••  xn = Sum{m=0:n}[S(n,m).x(m)] 
m = 6, n = 6,7,8,...   1, 21, 266, 2646, 22827, 179487, •••  See also OEIS A008277 
m = 7, n = 7,8,9,...   1, 28, 462, 5880, 63987, 627396, •••   
m = 8, n = 8,9,10,...   1, 36, 750, 11880, 159027, 1899612, •••   
m = 9, n = 9,10,11,...   1, 45, 1155, 22275, 359502, 5135130, •••   
More counting (enumeration) of various objects 
Subsets of a set of n labelled elements  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, •••  2n, for n=0,1,2,... 
Composition numbers  1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...  1 for n=0, else 2^(n-1); Distinct compositions of number n=0,1,2,... 
Catalan numbers C(n)  1, 1, 2, 5, 14, 42, 132, 429,1430,4862, •••  C(n)=C(2n,n)/(n+1); ubiquitous in number theory 
Bell numbers B(n)  1, 1, 2, 5, 15, 52, 203, 877, 4140, •••  Partitions of a set of n=0,1,2,... labelled elements 
Ordered Bell numbers or Fubini numbers  1, 1, 3, 13, 75, 541, 4683, 47293, •••  Weakly ordered partitions of n=0,1,2,... labelled elements 
Partition numbers p(n)  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, •••  Partitions of a set of n=0,1,2,... unlabelled elements 
Free labelled trees  1, 1, 3, 16, 125, 1296,16807,262144, •••  nn-2 (Cayley formula), n=1,2,3,... labelled vertices 
Rooted labelled trees  1, 2, 9, 64, 625, 7776, 117649, •••  nn-1, n=1,2,3,... labelled vertices 
Free unlabelled trees  1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235 •••  for n=1,2,3,... unlabeled vertices 
Rooted unlabelled trees  1, 2, 4, 9, 20, 48, 115, 286, 719, •••  for n=1,2,3,... unlabelled vertices 
Simple connected graphs  1, 1, 2, 6, 21, 112, 853, 11117, 261080, •••  for n = 1,2,3,... unlabelled nodes 
Compositions of powers of integer numbers 
Pythagorean triples (a,b,c), a2 + b2 = c2  (3,4,5) (5,12,13) (7,24,25) (8,15,17)  (9,40,41) (11,60,61) (12,35,37) (13,84,85) (16,63,65) ... 
Pythagorean quadruples, a2 + b2 + c2 = d2  (1,2,2,3) (2,3,6,7) (4,4,7,9) (1,4,8,9)  (6,6,7,11) (2,6,9,11) (10,10,23,27) (7,14,22,23) ... 
Pythagorean quintuples  (1,2,4,10,11) (1,2,8,10,13) ...  etc; there is an infinity of them in each category 
Taxicab numbers Ta(1), Ta(2), Ta(3), Ta(4),
     Ta(5),
     Ta(6) 
2, 1729, 87539319, 6963472309248,
48988659276962496,
24153319581254312065344, ••• 
Ta(n) is the smallest T(n) number, one that can be written as a
sum of two positive cubes in n different ways. Only six are known
and only Ta(1), Ta(2) are cubefree (not divisible by a cube). 
Hardy-Ramanujan number  1729 = 13+123 = 93+103  Smallest cubefree taxicab number T(2) 
Vojta's number  15170835645  Smallest cubefree T(3) number (see the link) 
Gascoigne's number  1801049058342701083   Smallest cubefree T(4) number (see the link) 
Special numbers related to divisors; σ(n) is the divisor function (sum of all divisors of n), and σ(n)-n is the sum of all proper divisors of n 
Perfect numbers  6, 28, 496, 8128, 33550336, •••  n = equals the sum of its own proper divisors = σ(n) - n 
Abundant numbers  12, 18, 20, 24, 30, 36, 40, 42, 48, 54, •••  n exceeds the sum of its own proper divisors; n > σ(n) - n 
Deficient numbers  1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, •••  n is less than the sum of its own proper divisors; n < σ(n) - n 
Odd abundant numbers  945, 1575, 2205, 2835, 3465, 4095, •••  Funny the smallest one is so large 
Odd abundant numbers not divisible by 3  5391411025, 26957055125, •••  see also 
Last integer not composed of two abundants  20161  Exactly 1456 integers are the sum of two abundants 
Amicable number pairs  (220,284); (1184,1210); (2620,2924); •••  n = σ(m) - m, m = σ(n) - n 
Superperfect numbers  2, 4, 16, 64, 4096, 65536, 262144, •••  n = σ(σ(n)) - n 
Special numbers related to absolute and/or relative primes, natural number factorizations, etc 
Prime numbers  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, •••  A prime number is divisible only by 1 and itself; excluding 1 
Twin prime numbers  3, 5, 11, 17, 29, 41, 59, 71, 101, 107, •••  For each prime p in this list, p+2 is also a prime 
Euler's totient function Φ(n) for n=1,2,3,...  1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, •••  Number of k's smaller than n and relatively prime to it 
Möbius function μ(n) for n=1,2,3,...  1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, •••  μ(n) = (-1)^k if n has k different prime factors; else μ(n) = 0 
Mersenne primes (first 7 for p = 2,3,5,7,13,17,19)  3, 7, 31, 127, 8191, 131071, 524287, •••  Some M(p) = 2p-1; p prime; Largest known: M(43112609) 
Pseudoprimes to base 2 (Sarrus numbers)  341, 561, 645, 1105, 1387, 1729, •••  Composite odd n such that 2n-1 = 1 (mod n) 
Pseudoprimes to base 3  91, 121, 286, 671, 703, 949, 1105, •••  Composite odd n such that 3n-1 = 1 (mod n) 
Carmichael's pseudoprimes  561, 1105, 1729, 2465, 2821, 6601, •••  Composite odd n such that bn-1 = 1 (mod n) for any coprime b 
Euler's pseudoprimes in base 2  341, 561, 1105, 1729, 1905, 2047, •••  Composite odd n such that 2(n-1)/2 = ±1 (mod n) 
Ishango bone prime quadruplet  11, 13, 17, 19   Crafted in the paleolithic Ishango bone 
Named sequences of binary numbers {0,1} or {-1,+1}; n-th term often refers to the digits di of the binary expansion B(n) of n. 
Baum - Sweet sequence  1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,1,0,0 •••  1 if B(n) contains no block of 0's of odd length 
Fibonacci words  0, 01, 010, 01001, 01001010, ...  Like Fibonacci recurrence, but using string concatenation 
Infinite Fibonacci word  010010100100101001010010010 •••  Infinite continuation of the above 
Golay - Rudin - Shapiro sequence  +1,+1,+1,-1,+1,+1,-1,+1,+1,+1,+1,-1 •••  (-1)^Sum[i]{didi+1} 
Thue - Morse sequence  0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0 •••  1 if B(n) contains an odd number of ones (parity = 1) 
Other named sequences of integers 
Euler numbers E(n) for n=0,2,4,...  1, -1, 5, -61, 1385, -50521, 2702765, •••  1/cosh(t) = Sum[n=0,∞]{tn(E(n)/n!)}; the odd ones are 0 
Fibonacci numbers F(n)  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, •••  Fn = Fn-1 + Fn-2; F0 = 0, F1 = 1 
Golomb's sequence (or Silverman's sequence)  1, 2,2, 3,3, 4,4,4, 5,5,5, 6,6,6,6, 7,7,7 •••  a(1)=1, a(n) is the (least possible) number of times n occurs 
Heegner numbers h  1, 2, 3, 7, 11, 19, 43, 67, 163 (full set)  The quadratic ring Q(√(-h)) has class number 1 
Lucas numbers L(n)  2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, •••  Fn = Fn-1 + Fn-2; F0 = 2, F1 = 1 
Sylvester's (or Ahmes') sequence  2, 3, 7, 43, 1807, 3263443, •••  sn is the product of previous members, plus 1 
Selected rational numbers and their sequences 
Bernoulli numbers B0 = 1, B1 = -1/2, B2k+1= 0 for k>1, Bm = δm,0 - Sum{k=0...(m-1)}[C(m,k)Bk/(m-k+1)]; x/(ex-1) = Sum{k=0,...}[Bnxn/n!]; Example: B10 = 5/66 
Bn = N/D; n = 2,4,6,...  N: 1,-1, 1, -1, 5, -691, 7, -3617, 43867, •••  D: 6, 30, 42, 30, 66, 2730, 6, 510, 798, ••• 
Other 
Rationals ≤ 1, sorted by denominators:nominators  1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, •••  Take the inverse values for rationals ≥ 1 
Farey sequences of fractions Fn (example n=5)  0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1, ...  F1 = { 0/1,1/1 }; for higher orders, interpolate ( a/c,b/d ) → a+b/c+d 
Stern - Brocot sequence (example n=4)  1/1, 1/2, 2/1, 1/3, 2/3, 3/2, 3/1, 1/4, 2/5, 3/5, 3/4, ...  Wraps up the binary Stern - Brocot tree 
Useful function-related values and constants 
Function gamma Γ(x) = Intg[0,x]{tx-1e-t dt}; Γ(x+1) = xΓ(x); Γ(1) = Γ(2) = 1 
Γ(x) minimum location; x ≥ 0  1.461 632 144 968 362 341 262 659 •••  Also the positive root of digamma function ψ(x) 
Γ(x) minimum value; x ≥ 0  0.885 603 194 410 888 700 278 815 •••  For x > 0, the Gamma function minimum is unique 
Γ(1/2)  1.772 453 850 905 516 027 298 167 •••  √π, crops up very often 
Γ(1/3)  2.678 938 534 707 747 633 655 692 •••  Γ(2/3) = 1.354 117 939 426 400 416 945 288 ••• 
Γ(1/4)  3.625 609 908 221 908 311 930 685 •••  Γ(3/4) = 1.225 416 702 465 177 645 129 098 ••• 
Γ(i)    (real and imaginary parts)  -0.154 949 828 301 810 685 124 955 ...  - i 0.498 015 668 118 356 042 713 691 ... 
Function digamma ψ(x) = (dΓ(x)/dx)/Γ(x); ψ(x+1) = ψ(x) + 1/x; For positive root, see above. 
ψ(1) = -γ  - 0.577 215 664 901 532 860 606 512 •••  ψ(2) = 1-γ = +0.422 784 335 098 467 139 393 488 ••• 
ψ(1/2) = -γ-2ln(2)  - 1.963 510 026 021 423 479 440 976 •••  ψ(-1/2) = 0.036 489 973 978 576 520 559 024 ... 
Riemann zeta function ζ(x) = Sum[k=0,1,2,...]{k^(-x)} and analytic continuations which exist everywhere except at x = 1 
ζ(-1/2) = -ζ(3/2)/(4π)  -0.207 886 224 977 354 566 017 306 •••  ζ(0) = -0.5 exact; ζ(-1) = -1/12 exact 
ζ(1/2)  -1.460 354 508 809 586 812 889 499 •••  ζ(3/2) = 2.612 375 348 685 488 343 348 567 ••• 
ζ(2)   = π2 /6  1.644 934 066 848 226 436 472 415 •••  ζ(3)   = 1.202 056 903 159 594 285 399 738 •••, see Apéry's 
ζ(4)   = π4 /90  1.082 323 233 711 138 191 516 003 •••  ζ(5)   = 1.036 927 755 143 369 926 331 365 ••• 
ζ(6)   = π6 /945  1.017 343 061 984 449 139 714 517 •••  ζ(7)   = 1.008 349 277 381 922 826 839 797 ••• 
ζ(8)   = π8 /9450  1.004 077 356 197 944 339 378 685 •••  ζ(9)   = 1.002 008 392 826 082 214 417 852 ••• 
ζ(10) = π10 /93555  1.000 994 575 127 818 085 337 145 •••  ζ(11) = 1.000 494 188 604 119 464 558 702 ••• 
ζ(12) = π12 691/638512875  1.000 246 086 553 308 048 298 637 •••  ζ(13) = 1.000 122 713 347 578 489 146 751 ••• 
ζ(i)    (real and imaginary parts)  0.003 300 223 685 324 102 874 217 •••  -i 0.418 155 449 141 321 676 689 274 ••• 
1st root, imaginary part  14.134 725 141 734 693 790 457 251 •••  Note: all listed roots have real parts +0.5 
2nd root  21.022 039 638 771 554 992 628 479 •••  3rd root: 25.010 857 580 145 688 763 213 790 ••• 
4th root; for more, see OEIS Wiki  30.424 876 125 859 513 210 311 897 •••  5th root: 32.935 061 587 739 189 690 662 368 ••• 
Dawson integral F(x) = e-x^2 Intg[0,x]{e-t^2 dt} 
F(x) maximum: Location xmax  0.924 138 873 004 591 767 012 823 •••  F(x) being an odd function; there is a minimum at -xmax 
F(x) maximum: Value  0.541 044 224 635 181 698 472 759 •••  F(xmax) = 1/(2xmax). 
F(x) inflection: Location xi  1.501 975 268 268 611 498 860 348 •••  Dawson integral: see above. 
F(x) inflection: Value at xi  0.427 686 616 017 928 797 406 755 ...  F(xi) = xi/(2xi2-1). 
Exponential integral Ei(x) = Intg[1,∞]{exp(-xt)/t dt} and Logarithmic integral Li(x) = Intg[0,x]{dt/ln(t)}, x ≥ 0 (use principal value when crossing t = 1) 
Real root of Li(x)  1.451 369 234 883 381 050 283 968 •••  μ; this is the Soldner's constant 
Li(2)  1.045 163 780 117 492 784 844 588 •••  Li(0) = Li(μ) = 0; Li(1) = -∞; Li(∞) = ∞ 
Real root of Ei(x)  0.372 507 410 781 366 634 461 991 •••  ln(μ) 
Ei(1)  1.895 117 816 355 936 755 466 520 •••  Ei(0) = -∞; Ei(ln(μ)) = 0; Ei(-∞) = 0; Ei(∞) = ∞ 
Ei(-1)  - 0.219 383 934 395 520 273 677 163 •••  Equals -(Gompertz constant)/e ... 
Function y(x) = xx = exp(x.ln(x)), x ≥ 0: 
Location of minimum  0.367 879 441 171 442 321 595 523 •••  xmin = 1/e. The minimum is unique. 
Value at minimum  0.692 200 627 555 346 353 865 421 •••  equals e-1/e. 
Function sinc(x) = sin(x)/x and its Hilbert transform Hsinc(x) = [1-cos(x)]/x, appearing in spectral theory (transient data truncation artifacts) 
sinc(x):    Half-height argument  1.895 494 267 033 980 947 144 035 •••  Solution of sinc(x) = 1/2 
sinc(x):    First minimum location  4.493 409 457 909 064 175 307 880 •••  A solution of tan(x) = x 
sinc(x):    First minimum value  -0.217 233 628 211 221 657 408 279 •••   
Hsinc(x): First maximum location  2.331 122 370 414 422 613 667 835 ...  A solution of x.sin(x) = 1-cos(x) 
Hsinc(x): First maximum value  0.724 611 353 776 708 475 738 990 ...   
Geometry constants 
Magic angle φm and Tetrahedral angle θm. Notes: φm = acos(1/√3) = atan(√2) = π/2 - asin(1/√3); π-θm = acos(1/3) = atan(2√2) 
Magic angle φm  0.955 316 618 124 509 278 163 857 •••  Degrees: 54.735 610 317 245 345 684 622 999 ... 
Complementary magic angle φ'm = π/2 - φm  0.615 479 708 670 387 341 067 464 •••  Degrees: 35.264 389 682 754 654 315 377 000 ... 
Tetrahedral angle θm = 2φm  1.910 633 236 249 018 556 327 714 •••  Degrees: 109.471 220 634 490 691 369 245 999 ... 
Complemetary tetrahedral angle θ'm = π - θm  1.230 959 417 340 774 682 134 929 •••  Degrees: 70.528 779 365 509 308 630 754 000 ... 
Sphere, the Queen of all bodies 
Volume / radius3 4.188 790 204 786 390 984 616 857 •••  4π/3; volume of a sphere with unit radius 
Radius / Volume1/3 0.620 350 490 899 400 016 668 006 •••  (3/(4π))1/3; radius of a sphere with unit volume 
Surface / radius2 12.566 370 614 359 172 953 850 573 •••  4π; surface of a sphere with unit radius 
Radius / Surface1/2 0.282 094 791 773 878 143 474 039 •••  1/(4π)1/2; radius of a sphere with unit surface 
Platonic solids: Tetrahedron, regular, 4 vertices, 6 edges, 4 faces, 3 edges/vertex, 3 edges/face, 3 faces/vertex. 
Volume / edge3 0.117 851 130 197 757 920 733 474 •••  (√2)/12 
Surface / edge2 1.732 050 807 568 877 293 527 446 •••  √3 
Height / edge  0.816 496 580 927 726 032 732 428 •••  (√6)/3 
Angle between an edge and a face  0.955 316 618 124 509 278 163 857 •••  magic angle φm (see above) 
Dihedral angle (between adjacent faces)  1.230 959 417 340 774 682 134 929 ...  complementary tetrahedral angle θ'm (see above) 
Tetrahedral angle (vertex-center-vertex)  1.910 633 236 249 018 556 327 714 ...  θm (see above) 
Circumscribed sphere radius / edge  0.612 372 435 695 794 524 549 321 ...  Circumradius = (√6)/4, congruent with vertices 
Midsphere radius / edge  0.353 553 390 593 273 762 200 422 ...  Midradius = 1/√8, tangent to edges 
Inscribed sphere radius / edge  0.204 124 145 231 931 508 183 107 ...  Inradius = (√6)/12, tangent to faces; Circumradius/Inradius = 3 
Vertex solid angle  0.551 285 598 432 530 807 942 144 ...  acos(23/27) steradians 
Polar angle of circumscribed cone  0.615 479 708 670 387 341 067 464 ...  complementary magic angle φ'm (see above) 
Solid angle of circumscribed cone  1.152 985 986 532 130 094 749 141 ...  2π(1-sqrt(2/3)) steradians 
Platonic solids: Octahedron, regular, 6 vertices, 12 edges, 8 faces, 4 edges/vertex, 3 edges/face, 4 faces/vertex. 
Volume / edge3 0.471 404 520 791 031 682 933 896 ...  (√2)/3 
Surface / edge2 3.464 101 615 137 754 587 054 892 ...  2√3 
Dihedral angle (between adjacent faces)  1.910 633 236 249 018 556 327 714 ...  tetrahedral angle (see above) 
Circumscribed sphere radius / edge  0.707 106 781 186 547 524 400 844 •••  Circumradius = 1/√2, congruent with vertices 
Midsphere radius / edge  0.5 exact  Midradius, tangent to edges 
Inscribed sphere radius / edge  0.408 248 290 463 863 016 366 214 ...  1/√6; Circumradius/Inradius = √3 
Vertex solid angle  1.359 347 637 816 487 748 385 570 ...  4asin(1/3) steradians 
Polar angle of circumscribed cone  0.785 398 163 397 448 309 615 660 ...  π/4; Degrees: 45 exact 
Solid angle of circumscribed cone  1.840 302 369 021 220 229 909 405 ...  2π(1-sqrt(1/2)) steradians 
Platonic solids: Cube, or Hexahedron, 8 vertices, 12 edges, 6 faces, 3 edges/vertex, 4 edges/face, 3 faces/vertex: 
Body diagonal / edge  1.732 050 807 568 877 293 527 446 ...  √3. Diagonal of a cube with unit side 
Body diagonal / Face diagonal  1.224 744 871 391 589 049 098 642 ...  sqrt(3/2) 
Angle between body diagonal and an edge  0.955 316 618 124 509 278 163 857 ...  magic angle φm (see above) 
Angle between body and face diagonals  0.615 479 708 670 387 341 067 464 ...  complementary magic angle φ'm (see above) 
Circumscribed sphere radius / edge  0.866 025 403 784 438 646 763 723 ...  Circumradius = (√3)/2, congruent with vertices 
Midsphere radius / edge  0.707 106 781 186 547 524 400 844 •••  Midradius = 1/√2, tangent to edges 
Inscribed sphere radius / edge  0.5 exact  Circumradius/Inradius = √3 
Vertex solid angle  1.570 796 326 794 896 619 231 321 ...  π/2 steradians 
Polar angle of circumscribed cone  0.955 316 618 124 509 278 163 857 ...  magic angle φm (see above) 
Solid angle of circumscribed cone  2.655 586 578 711 150 775 737 130 ...  2π(1-sqrt(1/3)) steradians 
Platonic solids: Icosahedron, regular, 12 vertices, 30 edges, 20 faces, 5 edges/vertex, 3 edges/face, 5 faces/vertex. 
Volume / edge3 2.181 694 990 624 912 373 503 822 ...  2/6 = 5(3 + √5)/12, where Φ is the golden ratio 
Surface / edge2 8.660 254 037 844 386 467 637 231 ...  5√3 
Dihedral angle (between adjacent faces)  2.411 864 997 362 826 875 007 846 ...  2atan(Φ2); Degrees: 138.189 685 104 221 401 934 142 083 ... 
Circumscribed sphere radius / edge  0.951 056 516 295 153 572 116 439 ...  ξΦ/2 = (sqrt(10)+2sqrt(5))/4, ξ being the associate of Φ 
Midsphere radius / edge  0.809 016 994 374 947 424 102 293 ...  Midradius = Φ/2, tangent to edges 
Inscribed sphere radius / edge  0.755 761 314 076 170 730 480 133 ...  Inradius = Φ2/(2√3) = sqrt(42+18√5)/12 
Vertex solid angle  2.634 547 026 044 754 659 651 303 ...  2π - 5asin(2/3) steradians 
Polar angle of circumscribed cone  1.017 221 967 897 851 367 722 788 ...  acos(sqrt((5-√5)/10)); Degrees: 58.282 525 588 538 994 675 ... 
Solid angle of circumscribed cone  2.979 919 307 985 462 371 739 387 ...  2π(1-sqrt((5-√5)/10)) steradians 
Platonic solids: Dodecahedron, regular, 20 vertices, 30 edges, 12 faces, 3 edges/vertex, 5 edges/face, 3 faces/vertex. 
Volume / edge3 7.663 118 960 624 631 968 716 053 ...  (5Φ3)/(2ξ2) = (15+7√5)/4 
Surface / edge2 20.645 728 807 067 603 073 108 143 ...  15Φ/ξ = 3.sqrt(25+10√5) 
Dihedral angle (between adjacent faces)  2.034 443 935 795 702 735 445 577 ...  2atan(Φ); Degrees: 116.565 051 177 077 989 351 572 193 ... 
Circumscribed sphere radius / edge  1.401 258 538 444 073 544 676 677 ...  Circumradius = Φ(√3)/2 = (sqrt(15)+sqrt(3))/4 
Midsphere radius / edge  1.309 016 994 374 947 424 102 293 ...  Midradius = Φ2/2, tangent to edges 
Inscribed sphere radius / edge  1.113 516 364 411 606 735 194 375 ...  Inradius = Φ2/(2ξ) = sqrt(250+110√5)/20 
Vertex solid angle  2.961 739 153 797 314 967 874 090 ...  π - atan(2/11) steradians 
Polar angle of circumscribed cone  1.311 325 654 302 976 339 315 817 ...  acos(sqrt(1-Φ/√3)); Degrees: 75.133 425 558 791 741 514 ... 
Solid angle of circumscribed cone  4.671 114 867 409 770 916 675 651 ...  2π(1-sqrt(1-Φ/√3)) steradians 
Surface-to-Volume indices for CLOSED 3D bodies: σ3 = Surface/Volume2/3, sorted by value: 
Sphere  4.835 975 862 049 408 922 150 900 ...  (36π)1/3; the absolute minimum for closed bodies 
Icosahedron, regular  5.148 348 556 199 515 646 330 812 ...  (5√3)/[5(3+√5)/12]2/3; Platonic icosahedron 
Dodecahedron, regular  5.311 613 997 069 083 669 796 666 ...  (3√(25+10√5))/[(15+7√5)/4]2/3; Platonic dodecahedron 
Cylinder, closed, with minimum σ3  5.535 810 445 932 085 257 290 411 ...  (54π)1/3; attained for Height = Diameter 
Octahedron, regular  5.719 105 757 981 619 442 544 453 ...  (2√3)/[(√2)/3]2/3; Platonic octahedron 
Cube  6  exact 
Cone (closed) with minimum σ3  6.092 947 785 379 555 603 436 316 ...  (72π)1/3; attained for Height = (Base diameter)√2 
Tetrahedron, regular  7.205 621 731 056 016 360 052 792 ...  (√3)/[(√2)/12]2/3; Platonic tetrahedron 
Surface-to-Volume indices for OPEN 3D bodies: σ3 = Outer_Surface/Volume2/3: 
Cup (half-closed cylinder) with minimum σ3  4.393 775 662 684 569 789 060 427 ...  1/3; attained for Height = Radius 
Tube (open cylinder) for length = diameter = 1  3.690 540 297 288 056 838 193 607 ...  (16π)1/3; scales as (Length/Diameter)1/3 
Cone (open) with minimum σ3  4.188 077 948 623 138 128 725 597 ...  ((√3)27π/2)1/3; attained for Height = (Base radius)√2 
Solid angle fractions f cut-out by cones with a given polar angle θ, f = (1 - cosθ)/2; the subtended solid angle is 4πf 
θ = θ'm, the complementary tetrahedral angle  0.333 333 333 333 333 333 333 333 ...  1/3 exact 
θ = 60 degrees  0.25  1/4 exact 
θ = 1 radian  0.229 848 847 065 930 141 299 531 ...  (1-cos(1))/2 
θ = φm, the magic angle  0.211 324 865 405 187 117 745 425 ...  (1-sqrt(1/3))/2; also the Knuth's constant 
θ = 45 degrees  0.146 446 609 406 726 237 799 577 ...  (1-1/√2)/2 
θ = φ'm, the complementary magic angle  0.091 751 709 536 136 983 633 785 ...  (1-sqrt(2/3))/2 
θ = 30 degrees  0.066 987 298 107 780 676 618 138 ...  (1-sqrt(3/4))/2 
θ = 15 degrees  0.017 037 086 855 465 856 625 128 ...  (1-sqrt((1+sqrt(3/4))/2))/2 
θ = 0.5 degrees (base of 1 degree diameter)  1.903 846 791 435 563 132 241 ...e-5  Steradians: 2.392 444 437 413 785 769 530 ...e-4 
Polar angles θ of cones cutting a given fraction f of the full solid angle, θ = acos(1-2f) 
f = (Φ-1)/Φ, where Φ is the golden-ratio  1.332 478 864 985 030 510 208 009 ...  Degrees: 76.345 415 254 024 494 986 936 602 ... 
f = 1/3  1.230 959 417 340 774 682 134 929 ...  Degrees: 70.528 779 365 509 308 630 754 000 ... 
f = 1/4  1.047 197 551 196 597 746 154 214 ...  Degrees: 60 
f = 0.01 ( 1%)  0.200 334 842 323 119 592 691 046 ...  Degrees: 11.478 340 954 533 572 625 029 817 ... 
f = 1e-6 ( 1 ppm)  0.002 000 000 333 333 483 333 422 ...  Degrees: 0.114 591 578 124 766 407 153 079 ... 
Various solid angles in (ste)radians 
Square on a sphere with 1 degree sides  3.046 096 875 119 366 637 825 ...e-4  4 asin(sin(α/2)sin(β/2)); α = β = 1 degree = π/180 
Triangle on a sphere with 1 degree sides  1.319 082 346 912 923 487 761 ...e-4  See Huilier's formula 
Packing ratios (monodispersed) 
ρ2, 2D disks, thickest  0.906 899 682 117 089 252 970 392 ...  ρ2 = π / 2√3. Best covering of infinite 2D plane. 
2D disks, closest random  0.772 ± 0.002  Empirical & theoretical 
ρ3, 3D spheres, thickest  0.740 480 489 693 061 041 169 313 ...  ρ3 = π / 3√2. Best covering of infinite 3D space 
3D spheres, closest random  0.634 ± 0.007  Empirical & theoretical; in practice: a vibrated bed 
Perimeters of ellipses with major semi-axis 1, and minor semi-axis b (area = πab). Special cases: b=0 ... flat ellipse, b = 1 ... circle. 
b = 1/Φ, where Φ is the golden-ratio  5.154 273 178 025 879 962 492 835 ...  Golden ellipse 
b = 0.613 372 647 073 913 744 075 540 ...  π+2 = mean of flat ellipse and circle  Mid-girth ellipse differs from golden ellipse by < 1% 
b = 1/√2  5.402 575 524 190 702 010 080 698 ...  Balanced ellipse (interfocal_distance = minor_axis) 
b = 1/2 (Midway ellipse)  4.844 224 110 273 838 099 214 251 ...  b = 1/3: 4.454 964 406 851 752 743 376 500 ... 
b = 3/4  5.525 873 040 177 376 261 321 396 ...  b = 2/3: 5.288 479 863 096 863 263 777 221 ... 
b = 1/4  4.289 210 887 578 417 111 478 604 ...  b = 1/5: 4.202 008 907 937 800 188 939 832 ... 
b = 1/6  4.150 013 265 005 047 157 825 880 ...  b = 1/7: 4.116 311 284 366 438 220 003 847 ... 
b = 1/8  4.093 119 575 024 437 585 615 711 ...  b = 1/9: 4.076 424 191 956 689 482 335 178 ... 
b = 1/10  4.063 974 180 100 895 742 557 793 ...  b = 0.01: 4.001 098 329 722 651 860 747 464 ... 
b = 0.001  4.000 015 588 104 688 244 610 756 ...  b = 0.0001: 4.000 000 201 932 695 375 419 076 ... 
Moving sofa constants for largest sofa that can turn a unit-width hallway corner (it can't exceed 2√2) 
Gerver's  2.219 531 668 871 97 (?)  So far the largest 
Hammersley's  2.207 416 099 162 477 962 306 856 ...  π/2 + 2/π. This was a nice attempt 
Other, more or less named, geometry constants: 
Gravitoid constant  1.240 806 478 802 799 465 254 958 •••  2√(2/(3√3)). Width/Depth ratio of gravitoid curve and gravidome 
Minimum area of a constant-width figure  0.704 770 923 010 457 972 467 598 •••  (pi - sqrt(3))/2 for width = 1. See Reuleaux triangle 
Moser's worm constant  0.232 239 210 ••• ?  Area of smallest region accomodating any curve of length 1 
Square drill constant  0.987 700 390 736 053 460 131 999 •••  Portion of square area covered by a Reuleaux drill 
Universal parabolic constant  2.295 587 149 392 638 074 034 298 •••  ln(1+√2)+√2. Arc-to-latus_rectum ratio in any parabola. 
Area doubling (Pythagora's) constant  1.414 213 562 373 095 048 801 688 ...  √2. Area-doubling scale factor 
Area tripling (Theodorus') constant  1.732 050 807 568 877 293 527 446 ...  √3. Area-tripling scale factor 
Volume doubling (Delian's) constant  1.259 921 049 894 873 164 767 210 ...  21/3. Volume-doubling scale factor 
Volume tripling constant  1.442 249 570 307 408 382 321 638 ...  31/3. Volume-tripling scale factor 
Statistics and probability constants 
Normal probability distribution with density N(x,σ): 
Density maximum * σ  0.398 942 280 401 432 677 939 946 ...  1/√(2π), attained at x = 0 
Percentiles: x/σ for which Intg[-∞,x]{N(x,σ)dx} = P, Intg[-x,x]{N(x,σ)dx} = 2P-1 
75%  0.674 489 750 196 081 743 202 227 •••  Probable error: x/σ for which Intg[-x,x]{N(x,σ)dx} = 0.5 
80%  0.841 621 233 572 914 205 178 706 ...  85% ... 1.036 433 389 493 789 579 713 244... 
90%  1.281 551 565 544 600 466 965 103 ...  95% ... 1.644 853 626 951 472 714 863 848 ... 
98%  2.053 748 910 631 823 052 937 351 ...  99% ... 2.326 347 874 040 841 100 885 606 ... 
99.9%  3.090 232 306 167 813 541 540 399 ...  99.99% ... 3.719 016 485 455 680 564 393 660 ... 
Math constants useful in Sciences 
Planck's radiation law prl(x) = x3/(ex - 1), or prl(λ) = λ-5(e1/λ - 1)-1 
Root of (e-x + x/5 - 1) = 0  4.965 114 231 744 276 303 698 759 ...  Related to prl maximum. 
Integral of x3/(ex - 1) over [0,∞]  6.493 939 402 266 829 149 096 022 ...  π4/15; related to prl integral. 
Magic angle. Root of P2 = (1-3.cos2(φ))/2, P2(x) being the 2nd-order Legendre polynomial. 
φm  0.955 316 618 124 509 278 163 857 ...  acos(1/√3); Degrees: 54.735 610 317 245 345 684 622 999 ... 
φm complement (π/2 - φm)  0.615 479 708 670 387 341 067 464 ...  asin(1/√3); Degrees: 35.264 389 682 754 654 315 377 000 ... 
Ideal gas statistics with n particles per unit volume 
1st Chandrasekhar constant c = Γ(4/3)/(4π/3)1/3 0.553 960 278 365 090 204 701 121 ...  Mean distance to nearest neighbor = c/n1/3 
2nd Chandrasekhar constant C = (2π)-1/3 0.541 926 070 139 289 008 744 561 ...  Most probable distance to nearest neighbor = C/n1/3 
Spectral lines (peaks) of height H and half-height width W: 
Area of a Lorentzian peak / HW  1.570 796 326 794 896 619 231 321 ...  π / 2 
Area of a Gaussian peak / HW  1.064 467 019 431 226 179 315 267 ...  sqrt(π /(4ln2)) 
Area of a Sinc peak / HW  0.828 700 120 129 003 061 896 869 ...  π/(2η), η being defined by sinc(η) = 1/2 (see sinc function) 
The Bloembergen-Purcell-Pound function, bpp(x) = x/(1+x2) + 4x/(1+4x2), ubiquitous in the theory of 2nd rank relaxation processes 
bpp(x) maximum: Location xmax  0.615 795 146 961 756 244 755 982 ...  bpp(x) being an odd function; there is a minimum at -xmax 
bpp(x) maximum: Value at xmax  1.425 175 719 086 501 535 329 674 ...  For first term only: bpp1,max(y) = 0.5, for y = 1 
First roots ξn of sinc(n,x) for n = 0, 1, 2, 3, 4 (nD Fourier transform of an n-dimensional unit sphere): 
ξ0  2.404 825 557 695 772 768 621 631 ...  sinc(0,x) = J0(x), the Bessel function 
ξ1  3.141 592 653 589 793 238 462 643 ...  sinc(1,x) = sin(x)/x = sinc(x) = j0(x), 1st kind spherical Bessel 
ξ2  3.831 705 970 207 512 315 614 435 ...  sinc(2,x) = 2J1(x)/x 
ξ3 ,  also location of 1st negative lobe of sinc(1,x)  4.493 409 457 909 064 175 307 880 ...  sinc(3,x) = 3[sin(x)/x - cos(x)]/x2 = 3j1(x)/x 
ξ4  5.135 622 301 840 682 556 301 401 ...  sinc(4,x) = 8J2(x)/x3 
Engineering constants; click here for conventional physical constants instead 
Amplitude / Effective_Amplitude  1.414 213 562 373 095 048 801 688 ...  √2, holds only for harmonic functions 
Power factor of 2 (or 0.5) in dB  ±3.010 299 956 639 811 952 137 388 ...  ±10.log10(2); corresponding amplitudes ratio is √2 : 1 
Amplitude factor of 2 (or 0.5) in dB  ±6.020 599 913 279 623 904 274 777 ...  ±20.log10(2) 
±1 dB ratios: 
Power  1.258 925 411 794 167 210 423 954 ...  10+1/10
Inverse power  0.794 328 234 724 281 502 065 918 ...  10-1/10
Amplitude  1.122 018 454 301 963 435 591 038 ...  10+1/20
Inverse amplitude  0.891 250 938 133 745 529 953 108 ...  10-1/20
±3 dB ratios: 
Power  1.995 262 314 968 879 601 352 455 ...  10+3/10  +3 dB in power or +6 dB in amplitude
Inverse power  0.501 187 233 627 272 285 001 554 ...  10 -3/10  -3 dB in power or -6 dB in amplitude
Amplitude  1.412 537 544 622 754 302 155 607 ...  10+3/20
Inverse amplitude  0.707 945 784 384 137 910 802 214 ...  10-3/20
Music and acoustics: 
Half-note frequency ratio  1.059 463 094 359 295 264 561 825 •••  21/12 
"Perfect fifth" ratio  3/2, exact  also 2/3 
Pythagorean comma  1.013 643 264 770 507 8125  (3/2)12/27, frequency ratio of 12 perfect fifth to 7 octaves 
Computer and Software Engineering constants 
Decadic-to-binary precision/capacity factor  3.321 928 094 887 362 347 870 319 •••  ln2(10);  Example:  7 decadic digits require 23+ binary bits 
Binary-to-decadic precision/capacity factor  0.301 029 995 663 981 195 213 738 •••  log10(2); Example: 31 binary bits require 9+ decimal digits 
Unsigned integer data types maximum values (for signed integers see the 3rd column) 
byte       (8 bits)   2^8-1  255  signed max = 2^7-1 = +127 
word   (16 bits)   2^16-1  65'535  signed max = 2^15-1 = +32'767 
dword (32 bits)   2^32-1  4'294'967'295  signed max = 2^31-1 = +2'147'483'647 
qword (64 bits)   2^64-1  18'446'744'073'709'551'615  signed max = 2^63-1 = +9'223'372'036'854'775'807 
Bit configurations which can't be used as signed integers since, though formally negative, aritmetic negation returns the same value (weird numbers) 
8 bits  hex 0x80  signed -2^7 = -128 
16 bits  hex 0x8000  signed -2^15 = -32'768 
32 bits  hex 0x80000000  signed -2^31 = -2'147'483'648 
64 bits  hex 0x8000000000000000  signed -2^63 = -9'223'372'036'854'775'808 
Floating point (real) data types: 
float     (1+8+23 bits):   Maximum value  3.402823669209384634633746...e+38  2^(2^(8-1)); IEEE 754; bits are for: sign, exponent, mantissa 
float     (1+8+23 bits):   minimum value  1.401298464324817070923729...e-45  2*2^(-2^(8-1))*2^(-(23-1)) 
float     (1+8+23 bits):   epsilon value  1.1920928955078125 e-7  2^(-23), for x < ε, 1+x → 1 
double (1+11+52 bits): Maximum value  1.79769313486231590772930...e+308  2^(2^(11-1)); IEEE 754; bits are for: sign, exponent, mantissa 
double (1+11+52 bits): minimum value  4.94065645841246544176568...e-324  2*2^(-2^(11-1))*2^(-(52-1)) 
double (1+11+52 bits): epsilon value  2.220446049250313080847263...e-16  2^(-52), for x < ε, 1+x → 1 
Conversion constants 
1 rad (radian) in degrees  57.295 779 513 082 320 876 798 15 ...  180/π; planar angle; 57° 17' 44.806247...'' 
1° (degree) in radians  0.017 453 292 519 943 295 769 237 ...  π/180 
1 degree2 in sr (steradians)  0.000 304 617 419 786 708 599 346 ...  (π/180)2; used in astronomy; infinitesimal area limit
1 sr (steradian) in degree2  3282.806 350 011 743 794 781 694 ...  (180/π)2; inverse of the above; valid for infinitesimal areas
Full solid angle of 4π steradians in degrees2  41252.961 249 419 271 031 294 671 ...  4π/(π/180)2 = 3602

Notes

Truncated values
Real constants are truncated after the last listed digit, not rounded.
Complex-valued constants
are listed with their real part in the central column and imaginary part in the right column (otherwise reserved for notes).
Sequences and/or families of numeric constants
are edited in a manner I considered appropriate. However, suggestions are welcome.
Named constants attributions: the traditional and the new ones
In absolute majority of cases, the name of a constant was well established before my listing.
In a few cases, however, I have taken the liberty of assigning a name to an important constant.
I sincerely hope that these attributions will stick since they acknowledge merit. They include:
- Knuth's constant: the ratio c/m in congruence random-number generators of the type Xn+1 = (aXn+c) mod (m) which
  minimizes the correlation between successive values. See [Knuth 1997, Section 3.3.3, Equation 41].
- Chandrasekhar's constants (1st and 2nd): originally related to nearest-neighbor statistics in an ideal gas,
  but having a much more general significance for any 3D, uniformly random distributions of points.
- Blazy's constant: a recent prime-numbers generating constant which struck the imagination of many people.
Links to OEIS, the Online Encyclopedia of Integer Sequences
Bold dots after a value are links to the OEIS database (OEIS lists also decimal expansions of real-valued constants).
OEIS usually extends the values listed here and provides more references (I have in fact registered a few OEIS' myself).
This is work in progress; so far, not all listed constants and sequences have an OEIS link, even if it exists.
Warning: the OEIS decimal-point offset is not always what you expect).
Links: many links, other than those to OEIS and those appearing below,
are scattered through the text, accompanying the particular constants.
Feedback
- If you think a link is missing, please, let me know.
- I have received quite a few suggestions of constants that should be included; they are most appreciated.

References

  • Finch S.R.,
    Mathematical Constants,
    Cambridge University Press 2003. ISBN 0-521-81805-2. more >>
  • Hardy G.H, Wright E.M.,
    An Introduction to the Theory of Numbers,
    6th Edition, Oxford University Press 2009. ISBN 978-0199219865. more >>
  • Muller Jean-Michel, Brisebarre Nicolas, de Dinechin Florent, Jeannerod Claude-Pierre, Lefèvre Vincent, Melquiond Guillaume,
    Handbook of Floating-Point Arithmetic,
    Birkhäuser Boston 2009. ISBN 978-0817647049. more >>
  • Knuth D.E., The Art of Computer Programming
    Volume 1: Fundamental Algorithms. ISBN 0-201-89683-4
    Volume 2: Seminumerical Algorithms, ISBN 0-201-89684-2. See Section 3.3.3, Eq.41, for Knuth's constant.
    Volume 3: Sorting and Searching, ISBN 0-201-89685-0
    3rd Edition, Addison-Wesley 1997 (Vol.3, 1998).
  • Deo Narsingh,
    Graph Theory with Applications to Engineering and Computer Science,
    Prentice Hall 1974. ISBN 978-0133634730. more >>
  • Subramanian Chandrasekhar,
    Stochastic Problems in Physics and Astronomy,
    Reviews of Modern Physics 15, 1-89 (1943).


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Copyright ©2008 Stanislav Sýkora    DOI: 10.3247/SL2Math08.001 Designed by Stan Sýkora