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%I #12 Apr 10 2015 03:38:28
%S 11,29,37,43,53,61,97,109,137,173,199,227,257,269,271,313,347,353,379,
%T 397,401,409,421,433,439,509,521,577,599,601,617,641,643,647,659,677,
%U 733,761,773,809,821,827,839,863,911,919,929,937,941,947,953,971,1009
%N Primes p that divide A001008(k), the numerator of the k-th harmonic number H(k), for some k < p-1.
%C These primes are a subset of the non-harmonic primes A092102. Because these primes are analogous to the irregular primes A000928 that divide the numerators of Bernoulli numbers, they might be called H-irregular primes. The density of these primes is about 0.4 -- very close to the density of irregular primes.
%C These primes are called Harmonic irregular primes in the Wikipedia entry for "Regular prime" (see links). It may be noted that if p is known to be of this type and H(k) is the smallest Harmonic number divisible by p, then not only does k < p-1 hold, but k <= (p-1)/2. This is because, by symmetry, H(p-1-n) == H(n) (mod p), so that any eligible k lying between (p+1)/2 and p-1 would have a counterpart in the range between 1 and (p-1)/2. Furthermore, the minimal k cannot be exactly equal to (p-1)/2, because then p would be a Wieferich prime (A001220) and would also divide H(Int(p/4)). Thus k <= (p-3)/2, and this inequality is sharp because exact equality holds for p = 29, 37, 3373 (see A072984). - _John Blythe Dobson_, Apr 09 2015
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrregularPrime.html">Irregular Prime</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_prime">Regular prime</a>
%t n=2; Table[cnt=0; While[cnt==0, p=Prime[n]; k=1; h=0; While[cnt==0 && k<=(p-1)/2, h=h+1/k; If[Mod[Numerator[h], p]==0, cnt++ ]; k++ ]; n++ ]; p, {100}]
%Y Cf. A072984 (least k such that prime(n) divides A001008(k)).
%K nonn
%O 1,1
%A _T. D. Noe_, Feb 24 2004
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