Approximate chronological listing, with the most recent entries listed first
(the OEIS numbering itself is not chronological):
A213610 (Decimal expansion of the characteristic impedance of vacuum in SI units),
A213611 (Decimal expansion of the standard atmosphere in SI units),
A213612 (Decimal expansion of the duration of the Julian year in SI seconds),
A213613 (Decimal expansion of the duration of the Gregorian year in SI seconds),
A213614 (Decimal expansion of the length of one light year in meters),
Physics constants which were assigned immutable reference values in the SI system
General context: Within current metrological systems (SI + IAU definitions), several physics constants have been "assigned" immutable values. They thus became metrological reference points, no longer subject to experimental assessment. These may not be confused with "conventional" values of some empirical quantities (such as Josephson's constant) used in applied metrology, but not assigned, and therefore subject to possible future revisions. More ....
For OEIS it is proper to list the assigned metrological constants and some of their combinations, including:
speed of light (A003678) and its square (A182999), magnetic permeability of vacuum (A019694), electric permittivity of vacuum (A081799), standard gravity in (A072915), and the five constants listed above, all in basic SI units.
General discussion: Consider the 2^N numbers with N-digit binary expansion. Let a pair (v,w), here called a "transition", be such that there are exactly k+q digits which are '0' in v and '1' in w, and exactly k digits which are '1' in v and '0' in w. Then T(q;N,k) is the number of all such pairs. More ...
For given N and q, the rows of the triangle T(q;N,k) sum up to Sum[k]T(q;N,k) = C(2N,N-q) which is the total number of q-quantum transitions or, equivalently, the number of pairs in which the sum of binary digits of w exceeds that of v by exactly q.
The terminology stems from the mapping of the i-th digit onto quantum states of the i-th particle (-1/2 for digit '0', +1/2 for digit '1'), the numbers onto quantum states of the system, and the pairs onto quantum transitions between states. In magnetic resonance (NMR) the most intense transitions are the single-quantum ones (q=1) with k=0, called "main transitions", while those with k>0, called "combination transitions", tend to be weaker. Zero-, double- and, in general, q-quantum transitions are detectable by special techniques.
Specific cases: Each individual sequence covers q-quantum transitions for one value of q.
It lists the flattened triangle T(q;N,k), with rows N = 1,2,... and columns k = 0..floor((N-q)/2).
The case q=0 (zero-quantum transitions) is covered by the prior sequence A051288.
A213421 and A213422:
The real part and, respectively, the coefficient of i of Q^n, Q being the quaternion 2+i+j+k
Quaternions were so far a bit neglected in OEIS.
Decimal expansion of second Chandrasekhar's nearest-neighbor constant c
When n pointlike particles are distributed uniformly randomly in a unit volume, the most probable distance between any of them and its nearest neighbor is C/n^(1/3).
Decimal expansion of first Chandrasekhar's nearest-neighbor constant c
When n pointlike particles are distributed uniformly randomly in a unit volume, the mean distance between any of them and its nearest neighbor is c/n^(1/3).
Decimal expansion of the absolute minimum of sinc(x) = sin(x)/x (negated)
Minimum value of the first negative lobe of sinc(x), attained for |x| = A115365.
Decimal expansion of Brun's quadruple primes constant
Infinite sum of the inverse values of p, p+2, p+6, and p+8, where p ranges over all prime quadruplets (A007530). Current estimate is 0.8705883800. The 9th digit is probably 0.
Given integers n, p, q, 0 < p < q, the value of li(-n,-p/q) = SUM[k=0,1,2,...]((k^n)/(-p/q)^k), multiplied by ((p+q)^(n+1))/q, is the integer a(n) listed for n = 0,1,2,3,... In each sequence, the negative rational argument (-p/q) is kept constant (specified in the parentheses). The absolute values of the terms grow approximately exponentially but, for negative second arguments, they group into short blocks with alternating signs. More ....
Real and imaginary part of i-factorial (i!) and its absolute value and argument, respectively
Based on the indentities i! = gamma(1+i) = i*gamma(i).
Product of all primes in the interval ((n+1)/2,n]
a(n) = A034386(n)/A034386(1+floor((n+1)/2)). Notice that the interval is semi-open; a prime p is included in the product iff (n+1) < 2p ≤ 2n. The case n = 1 is special and set to 1 by convention.
Central binomial coefficient cb(n) purged of all primes exceeding (n+1)/2
A simple insight shows that the prime factors decomposition of cb(n) = binom(n,floor(n/2)) (i) does not contain any prime factor greater than n, (ii) contains exactly once all primes in the interval ((n+1)/2,n]. Hence, cb(n) is divisible by the product p2(n) of all primes in ((n+1)/2,n] (A212792). The relatively small elements of this sequence are a(n) = cb(n)/p2(n). For n>6, they can be shown to be void of any prime factor exceeding n/3.
Consider the set S of all b^n numbers which have n digits in base b. Define as "main transition" a pair (x,y) of elements of S such that x and y differ in base b in only one digit which in y exceeds that in x by 1. These sequences give the number of such transitions for the cases b = 3, 4, 5, 6, 7, 8, 9, 10. The case b=2 (S=1/2) is covered by the prior sequence A001787
The terminology originates from quantum theory of coupled spin systems (such as in magnetic resonance) with n particles, each with spin S = (b-1)/2. Then the i-th digit's value in base b can be intended as a label for the b = 2S+1 quantum states of the i-th particle. The most intense main quantum transitions then correspond to the above definition. Due to continuity, the correspondence holds regardless of how strongly coupled are the particles among themselves.
zeta(-1/2) = -zeta(3/2)/(4*Pi); zeta(3/2) being A078434
Decimal expansion of the argument of infinite power tower of i
This c, expressed in radiants, equals arg(z), where z is the complex solution of z = i^z or,
equivalently, z = i^i^i^...
Decimal expansion of the absolute value of infinite power tower of i
This c = |z|, where z is the complex solution of z = i^z or, eqivalently, z = i^i^i^...
A212436 and A212437:
Decimal expansions of the real and imaginary parts of e^(i/e).
Also cos(1/e) and sin(1/e), respectively.
Decimal expansion of the imaginary part of i^(1/4)
Also sin(Pi/8) or sine of 22.5 degrees.
The real part of i^(1/4), or cos(Pi/8), is the pre-existent A144981.
First a(n) > 1 which have the same sum of digits in all prime bases from 2 to p(n)
Here p(n) is the n-th prime (A000040). Conjecture: the sequence never terminates.
First a(n) > 1 such that its sum of digits is the same in base 10 as in base n
There might exist an n for which there is no solution, in which case a(n) would be set by convention to zero; however, no such case was found so far. Problem: does it exist?
First a(n) > 1 such that its sum of digits is the same in base 2 as in base n
Theoretically, there might exist an n for which there is no solution, in which case a(n) would be set by convention to zero; however, no such case was found so far. Problem: does it exist?
Decimal expansion of the only solution of x^(4/x^2) = x-1
Comes amazingly close to e (the relative error is smaller than 7.7*10^-6).
The sum of digits of a(n) in base b is the same for every prime b up to 11
This sequence is a subset of A135127 for bases 2,3,5,7, which is a subset of A135121 for bases 2,3,5, which is a subset of A037301 for bases 2,3. Problem: does it terminate?
Decimal expansion of (2/π)log(Φ), the exponential rate factor of golden spiral
In polar coordinates (r,θ), a golden spiral equation is r ~ e^(c*θ), where c is this constant,
with Φ being the golden ratio (A001622).
A212224: Decimal expansion of Φ^(2/π), the base of golden spiral
In polar coordinates (r,θ), a golden spiral equation is r ~ c^θ, where c is this constant,
with Φ being the golden ratio (A001622).
Decimal expansions of the real and imaginary parts of -(i^e), respectively
Also, respective decimal expansions of -cos(π*e/2) and -sin(π*e/2).
Integral of a sinc-shaped peak with unit height and unit half-height width
See first the general introduction in A211268.
This constant is the integral from -inf to +inf under a canonical sinc-shaped spectral peak S(x) defined by S(x) = sinc(2*x*eta), with sinc(z) = sin(z)/z, and eta=A199460, so that S(0) = 1 and S(+1/2) = S(-1/2) = 1/2.
Integral of a Gaussian peak with unit height and unit half-height width
In spectroscopy, when comparing absorbtion peak shapes, the functions are first scaled vertically and horizonatally to a canonical form with unit height and unit half-height width. The 4 most common canonical shapes are: rectangular R(x) = 1 for |x| ≤ 1/2 (0 otherwise), Lorentzian L(x) = 1/(1+(2x)^2), Gaussian G(x) = exp(-ln(2)(2x)^2), and sinc-type S(x) (see A211269). The areas A under such canonical peaks (integral from -inf to +inf) are 1.0 for R(x), (π/2)=A019669 for L(x), This constant for G(x), and A211269 for S(x). For a generic peak with height H and half-height width W belonging to the same canonical family, the area is A*H*W. Hence the practical importance of the constant A.
Decimal expansion of 2*sin(π/5); the 'associate' of the golden ratio
Golden ratio Φ is the real part of 2*exp(i*π/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron).
Numerators of the polylogarithm li(-n,-1/2)/2
Given an integer n, consider the series s(n) = li(-n,-1/2)) = SUM((-1)^k)(k^n)/2^k) for k=1,2,... Then s(n) = 2*a(n)/A131137(n+1).
Decimal expansion of the gravitoid constant
Ratio between the width and the depth of the gravitoid curve delimiting any axial section of a gravidome. A gravidome is an axially symmetric homogeneous body shaped in a way to produce, given a constant mass, the maximum possible gravitation field at a point (the barypole) on its surface. It is shaped like a tomato; with respect to a sphere it is somewhat flattened and the gravitoid constant describes the amount of the flattening.