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Decimal expansion of Pi^4/15.
7

%I #46 Nov 04 2024 02:30:07

%S 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,

%T 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,

%U 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

%N Decimal expansion of Pi^4/15.

%C 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.

%C 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.

%H Stanislav Sykora, <a href="/A231535/b231535.txt">Table of n, a(n) for n = 1..1000</a>

%H Ilham A. Aliev, Ayhan Dil, <a href="https://arxiv.org/abs/2008.02488">Tornheim-like series, harmonic numbers and zeta values</a>, arXiv:2008.02488 [math.NT], 2020, p. 4.

%H Stanislav Sykora, <a href="http://dx.doi.org/10.3247/SL1Math05.002">Surface Integrals over n-Dimensional Spheres</a>

%H Stanislav Sykora, <a href="http://dx.doi.org/10.3247/SL1Math05.001">Volume Integrals over n-Dimensional Ellipsoids</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Planck%27s_law">Planck's Law</a>

%H Ke Xiao, <a href="http://www.ejtp.com/articles/ejtpv8i25p379.pdf">Dimensionless Constants and Blackbody Radiation Laws</a>, Electronic Journal of Theoretical Physics, 8(2011), 379-388, Eq.6.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals 6*zeta(4), see A013662. - _Bruno Berselli_, Nov 12 2013

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

%F 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

%F From _Amiram Eldar_, Aug 14 2020: (Start)

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

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

%F Equals psi'''(1), the third derivative of the digamma function at 1. - _R. J. Mathar_, Aug 29 2023

%e 6.4939394022668291490960221792470074166485057115123614460978572926647...

%t RealDigits[Pi^4/15, 10, 105][[1]] (* _Bruno Berselli_, Nov 12 2013 *)

%o (PARI) Pi^4/15 \\ _Michel Marcus_, Aug 07 2020

%o (PARI) 6*prodeulerrat(p^4/(p^4-1)) \\ _Hugo Pfoertner_, Aug 07 2020

%Y Cf. A000796, A013662, A081819, A164109.

%K nonn,cons,easy,changed

%O 1,1

%A _Stanislav Sykora_, Nov 12 2013