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A099828
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Numerator of generalized harmonic number H(n,5). H(n,5) = Sum{1/k^5), k = 1..n. H(n,5) = 1, 33/32, 8051/7776, 257875/248832,... A099827 - H(n,5) multiplied by (n!)^5.
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22
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1, 33, 8051, 257875, 806108207, 268736069, 4516906311683, 144545256245731, 105375212839937899, 105376229094957931, 16971048697474072945481, 16971114472329088045481, 6301272372663207205033976933
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
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LINKS
| 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.
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FORMULA
| a(n) = Numerator[Sum[1/k^5, {k, 1, n}]] a(n) = Numerator[HarmonicNumber[n, 5]]
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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.
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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 (alex(AT)kolmogorov.com), Nov 07 2006
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CROSSREFS
| Cf. A099827 A001008, A007406, A007408, A007410.
Sequence in context: A114071 A057981 A183237 * A099827 A060705 A061687
Adjacent sequences: A099825 A099826 A099827 * A099829 A099830 A099831
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KEYWORD
| nonn,frac
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AUTHOR
| Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 27 2004
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