1,2

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 (alex(AT)kolmogorov.com), Nov 07 2006

Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 07 2006, Table of n, a(n) for n = 1..100

Eric Weisstein, The World of Mathematics: Wolstenholme's Theorem.

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

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

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

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

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.

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

Table[Numerator[Sum[1/k^5, {k, 1, n}]], {n, 1, 100}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 07 2006

Cf. A099827 A001008, A007406, A007408, A007410.

Sequence in context: A136541 A114071 A057981 this_sequence A099827 A060705 A061687

Adjacent sequences: A099825 A099826 A099827 this_sequence A099829 A099830 A099831

nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 27 2004