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A099828 Numerator of generalized harmonic number H(n,5). 24

%I

%S 1,33,8051,257875,806108207,268736069,4516906311683,144545256245731,

%T 105375212839937899,105376229094957931,16971048697474072945481,

%U 16971114472329088045481,6301272372663207205033976933

%N Numerator of generalized harmonic number H(n,5).

%C H(n,5) = Sum{1/k^5), k = 1..n.

%C a(n) is prime for n = {23, 25, 85, 147, 167, ...}. There is a Wolstenholme-like theorem: p divides a(p-1) for prime p. p^2 divides a(p-1) for prime p>7. p^3 divides a(p-1) for prime p = 5. p divides a((p-1)/2) for prime p = 37. p divides a((p-1)/3) for prime p = 37. p divides a((p-1)/4) for prime p = 37. p divides a((p-1)/5) for prime p = 11. p^2 divides a((p-1)/6) for prime p = 37. p divides a((p+1)/4) for prime p = 83. p divides a((p+1)/5) for prime p = 29. p divides a((p+1)/6) for prime p = 11. - _Alexander Adamchuk_, Nov 07 2006

%C See the Wolfdieter Lang link under A103345 on Zeta(k, n) = H(n, k) with the rationals for k=1..10, g.f.s and polygamma formulas. - _Wolfdieter Lang_, Dec 03 2013

%H Alexander Adamchuk, Nov 07 2006, <a href="/A099828/b099828.txt">Table of n, a(n) for n = 1..100</a>

%H Eric Weisstein, The World of Mathematics: <a href="http://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>.

%F a(n) = Numerator[Sum[1/k^5, {k, 1, n}]] a(n) = Numerator[HarmonicNumber[n, 5]]

%e a(2) = 1 + 1/2^5 = 33/32,

%e a(3) = 1 + 1/2^5 + 1/3^5 = 8051/7776.

%e H(n,5) = {1, 33/32, 8051/7776, 257875/248832, ... }.

%e Thus a(2) = Numerator[1 + 1/2^5] = Numerator[33/32] = 33, a(3) = Numerator[1 + 1/2^5 + 1/3^5] = Numerator[8051/7776] = 8051.

%t Numerator[Table[Sum[1/k^5, {k, 1, n}], {n, 1, 20}]] or Numerator[Table[HarmonicNumber[n, 5], {n, 1, 20}]]

%t Table[Numerator[Sum[1/k^5,{k,1,n}]],{n,1,100}] - _Alexander Adamchuk_, Nov 07 2006

%Y Denominators are A069052.

%Y A099827 = H(n,5) multiplied by (n!)^5.

%Y Cf. A001008, A007406, A007408, A007410.

%K nonn,frac

%O 1,2

%A _Alexander Adamchuk_, Oct 27 2004

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Last modified November 22 08:46 EST 2019. Contains 329389 sequences. (Running on oeis4.)