A007410 Numerator of Sum_{k=1..4} k^(-4).

1, 17, 1393, 22369, 14001361, 14011361, 33654237761, 538589354801, 43631884298881, 43635917056897, 638913789210188977, 638942263173398977, 18249420414596570742097, 18249859383918836502097, 18250192489014819937873

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• frac

Comments

p divides a(p-1) for prime p > 5. p divides a((p-1)/2) for prime p > 5. p^2 divides a((p-1)/2) for prime p = 31, 37. - Alexander Adamchuk, Jul 07 2006

p^2 divides a(p-1) for prime p = 37. - Alexander Adamchuk, Nov 07 2006

Denominators are A007480. See the W. Lang link under A103345 for the rationals and more.

The limit of the rationals Zeta(n) := Sum_{k=1..n} 1/k^4 as n -> infinity is (Pi^4)/90, which is approximately 1.082323234. See A013662.

References

• D. Y. Savio, E. A. Lamagna, and S.-M. Liu, Summation of harmonic numbers, pp. 12-20, in: E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..200
• Hisanori Mishima, Factorizations of many number sequences.
• Hisanori Mishima, Factorizations of many number sequences.

Crossrefs

Cf. A001008, A007406, A007408, A007480, A013662.

Programs

Mathematica

Numerator[Table[Sum[1/k^4,{k,1,n}],{n,1,20}]] (* Alexander Adamchuk, Jul 07 2006 *)
Accumulate[1/Range[20]^4]//Numerator (* Harvey P. Dale, Jun 28 2020 *)

PARI

a(n)=numerator(sum(k=1,n,1/k^4)) \\ Charles R Greathouse IV, Jul 19 2011

Formula

Numerators of the coefficients in the expansion of PolyLog(4, x)/(1 - x). - Ilya Gutkovskiy, Apr 10 2017