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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).

%I A093569 #7 Feb 16 2025 08:32:53
%S A093569 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,
%T A093569 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,
%U A093569 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
%N 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).
%C A093569 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.
%H A093569 T. D. Noe, <a href="/A093569/b093569.txt">Table of n, a(n) for n=1..10000</a>
%H A093569 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H A093569 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WieferichPrime.html">Wieferich Prime</a>
%e A093569 a(5) = 2 because 11 = prime(5) and there are 2 values, k = 3 and 7, such that 11 divides A001008(k).
%t A093569 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}]
%Y A093569 Cf. A001008, A001220, A092194.
%K A093569 nonn
%O A093569 1,5
%A A093569 _T. D. Noe_, Apr 01 2004