A231535 Decimal expansion of Pi^4/15.

6, 4, 9, 3, 9, 3, 9, 4, 0, 2, 2, 6, 6, 8, 2, 9, 1, 4, 9, 0, 9, 6, 0, 2, 2, 1, 7, 9, 2, 4, 7, 0, 0, 7, 4, 1, 6, 6, 4, 8, 5, 0, 5, 7, 1, 1, 5, 1, 2, 3, 6, 1, 4, 4, 6, 0, 9, 7, 8, 5, 7, 2, 9, 2, 6, 6, 4, 7, 2, 3, 6, 9, 7, 1, 2, 1, 8, 1, 3, 0, 7, 9, 3, 4, 1, 4, 5, 7, 8, 1, 5, 6, 5, 0, 1, 9, 9, 5, 0, 3, 3, 9, 7, 9, 4

Offset: 1

Author

Stanislav Sykora, Nov 12 2013

Keywords

• nonn
• cons
• easy

Comments

Under proper scaling, the radiation distribution density function in terms of frequency is given by prl(x) = x^3/(exp(x)-1), the Planck's radiation law. This constant is the integral of prl(x) from 0 to infinity and leads to the total amount of electromagnetic radiation emitted by a body.

Also, in an 8-dimensional unit-radius hypersphere, equals one-fifth of its surface (A164109), and twice the integral of r^2 over its volume.

Examples

			6.4939394022668291490960221792470074166485057115123614460978572926647...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..1000
• Ilham A. Aliev, Ayhan Dil, Tornheim-like series, harmonic numbers and zeta values, arXiv:2008.02488 [math.NT], 2020, p. 4.
• Stanislav Sykora, Surface Integrals over n-Dimensional Spheres
• Stanislav Sykora, Volume Integrals over n-Dimensional Ellipsoids
• Wikipedia, Planck's Law
• Ke Xiao, Dimensionless Constants and Blackbody Radiation Laws, Electronic Journal of Theoretical Physics, 8(2011), 379-388, Eq.6.
• Index entries for transcendental numbers

Crossrefs

Cf. A000796, A013662, A081819, A164109.

Programs

Mathematica

RealDigits[Pi^4/15, 10, 105][[1]] (* Bruno Berselli, Nov 12 2013 *)

PARI

Pi^4/15 \\ Michel Marcus, Aug 07 2020

PARI

6*prodeulerrat(p^4/(p^4-1)) \\ Hugo Pfoertner, Aug 07 2020

Formula

Equals 6*zeta(4), see A013662. - Bruno Berselli, Nov 12 2013

Equals Gamma(4)*zeta(4) = 2*3*Product_{prime p} (p^4/(p^4-1)). - Stanislav Sykora, Oct 20 2014

Equals Sum_{n, m >= 1} H(n+m)/(n*m*(n+m)) where H(n) is the n-th harmonic number. See Aliev and Dil. - Michel Marcus, Aug 07 2020

From Amiram Eldar, Aug 14 2020: (Start)

Equals Integral_{x=0..oo} x^3/(exp(x)-1) dx.

Equals Integral_{x=0..1} log(x)^3/(x-1) dx. (End)

Equals psi'''(1), the third derivative of the digamma function at 1. - R. J. Mathar, Aug 29 2023