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A125194 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. 0
41, 1599366601, 10877829357646990581304675244472669289, 100935935338172297894217692920950359818733561, 9217176064595104612826996436899733706027947436610177335077693637792069056822883934927465549747441 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=3..7.

Eric Weisstein's World of Mathematics, Harmonic Number

Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

FORMULA

a(n) = Numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,(Prime[n]-1)/2} ]] / Prime[n]^2 for n>2.

EXAMPLE

Prime[3] = 5.

a(3) = Numerator[ 1 + 1/2^10 ] / 5^2 = 1025 / 25 = 41.

MATHEMATICA

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}]

CROSSREFS

Cf. A120290, A119722, A001008, A007406, A007408, A007410.

Sequence in context: A297055 A228555 A297058 * A237639 A095189 A023932

Adjacent sequences:  A125191 A125192 A125193 * A125195 A125196 A125197

KEYWORD

frac,nonn

AUTHOR

Alexander Adamchuk, Jan 13 2007

STATUS

approved

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Last modified April 5 12:57 EDT 2020. Contains 333241 sequences. (Running on oeis4.)