Harmonic-Number-Related Divisibility

Leroy Quet qq-quet at mindspring.com
Sun Mar 7 03:58:53 CET 2004

Let H(m) be the m_th harmonic number:

H(m) = 1 + 1/2 + 1/3 + ...+ 1/m.

If we represent H(m) as a reduced fraction (and even if we do not reduce, 
for that matter...), I once proved that:

If p = odd prime, then

p divides (2*numerator(H(p-3)) -3*denominator(H(p-3))).

But I forgot now the specifics of the proof (or "proof"), which I hope is 

So, what about the sequence of

(2*numerator(H(p-3)) -3*denominator(H(p-3)))/p,

where the k_th term is for the k_th odd prime p?
(and we DO reduce)

0, 0, 2, 62,...

What can be said about this sequence, such as its factorization?

(Not in the EIS, but "2, 62" brings up a bunch of hits, which probably 
means very little.)



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