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A125194
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Numerator of generalized harmonic number H((p-1)/2,2p)= Sum[ 1/k^(2p), {k,1,(p-1)/2}] divided by p^2 for prime p>3.
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0
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41, 1599366601, 10877829357646990581304675244472669289, 100935935338172297894217692920950359818733561, 9217176064595104612826996436899733706027947436610177335077693637792069056822883934927465549747441
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OFFSET
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3,1
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COMMENTS
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Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3 (see A120290(n)). The numerator of generalized harmonic number H((p-1)/2,2p) is divisible by p^2 for prime p>3.
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LINKS
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FORMULA
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a(n) = Numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,(Prime[n]-1)/2} ]] / Prime[n]^2 for n>2.
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EXAMPLE
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Prime[3] = 5.
a(3) = Numerator[ 1 + 1/2^10 ] / 5^2 = 1025 / 25 = 41.
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MATHEMATICA
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Do[p=Prime[k]; f=0; Do[f=f+1/n^(2p); g=Numerator[f]; If[IntegerQ[g/(p)^2], Print[{p, g/p^2}]], {n, 1, (p-1)/2}], {k, 1, 100}]
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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