A093569 For p = prime(n), the number of integers k < p-1 such that p divides A001008(k), the numerator of the harmonic number H(k).

0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0

Offset

1,5

Comments

It is well-known that prime p >= 3 divides the numerator of H(p-1). For primes p in A092194, there are integers k < p-1 for which p divides the numerator of H(k). Interestingly, if p divides A001008(k) for k < p-1, then p divides A001008(p-k-1). Hence the terms of this sequence are usually even. The only exceptions are the two known Wieferich primes 1093 and 3511, A001220, which have 3 values of k < p-1 for which p divides A001008(k), one being k = (p-1)/2.

Links

T. D. Noe, Table of n, a(n) for n=1..10000

Eric Weisstein's World of Mathematics, Harmonic Number

Eric Weisstein's World of Mathematics, Wieferich Prime

Example

a(5) = 2 because 11 = prime(5) and there are 2 values, k = 3 and 7, such that 11 divides A001008(k).

Mathematica

len=500; Table[p=Prime[i]; cnt=0; k=1; While[k<p-1, If[Mod[Numerator[HarmonicNumber[k]], p]==0, cnt++ ]; k++ ]; cnt, {i, len}]

Crossrefs

Cf. A001008, A001220, A092194.

Sequence in context: A364213 A051907 A178176 * A073091 A125250 A048113

Adjacent sequences: A093566 A093567 A093568 * A093570 A093571 A093572

Keyword

nonn

Author

T. D. Noe, Apr 01 2004

Status

approved