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A099828 Numerator of generalized harmonic number H(n,5). 23
1, 33, 8051, 257875, 806108207, 268736069, 4516906311683, 144545256245731, 105375212839937899, 105376229094957931, 16971048697474072945481, 16971114472329088045481, 6301272372663207205033976933 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

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

LINKS

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

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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, Nov 07 2006

CROSSREFS

Denominators are A069052.

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

Cf. A001008, A007406, A007408, A007410.

Sequence in context: A057981 A219563 A183237 * A099827 A269793 A060705

Adjacent sequences:  A099825 A099826 A099827 * A099829 A099830 A099831

KEYWORD

nonn,frac

AUTHOR

Alexander Adamchuk, Oct 27 2004

STATUS

approved

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Last modified November 23 21:46 EST 2017. Contains 295141 sequences.