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A128820
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Numerator of alternating generalized harmonic number H'(p-1,2p) = Sum[ (-1)^(k+1)*1/k^(2*p), {k,1,p-1} ] ] divided by p^2 for prime p>2.
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0
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7, 2474315503, 53305712401979540402437, 5597916593064896381208777124641713285719656398067086247546781015747740847, 192635872080422175485338764164035657976855166649911323825254242037669356649787653784405726270977624462974729613783
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| Alternating generalized harmonic number is H'(n,m)= Sum[ (-1)^(k+1)*1/k^m, {k,1,n} ]. Numerator of H'(p-1,2n) is divisible by p for all integer n>0 and prime p>2. Numerator of H'(p-1,2p) is divisible by p^2 for prime p>2.
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LINKS
| Eric Weisstein, Link to a section of The World of Mathematics: Harmonic Number.
Eric Weisstein, Link to a section of The World of Mathematics: Wolstenholme's Theorem.
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FORMULA
| a(n) = Numerator[ Sum[ (-1)^(k+1)*1/k^(2*Prime[n]), {k,1,Prime[n]-1} ] ] / Prime[n]^2 for n>1.
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EXAMPLE
| Prime[2] = 3.
a(2) = numerator[ 1 - 1/2^6 ] / 3^2 = 63/9 = 7.
Prime[3] = 5.
a(3) = numerator[ 1 - 1/2^10 + 1/3^10 - 1/4^10 ] / 5^2 = 61857887575/25 = 2474315503.
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MATHEMATICA
| Table[ Numerator[ Sum[(-1)^(k+1)*1/k^(2*Prime[n]), {k, 1, Prime[n]-1} ] ] / Prime[n]^2, {n, 2, 10} ]
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CROSSREFS
| Cf. A119722 = Numerator of generalized harmonic number H(p-1, p)= Sum[ 1/k^p, {k, 1, p-1}] divided by p^3 for prime p>3. Cf. A001008, A119682, A120296.
Sequence in context: A075984 A109300 A124272 * A067485 A180225 A127886
Adjacent sequences: A128817 A128818 A128819 * A128821 A128822 A128823
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KEYWORD
| nonn
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AUTHOR
| Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 10 2007
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