

A092194


Primes p that divide A001008(k), the numerator of the kth harmonic number H(k), for some k < p1.


3



11, 29, 37, 43, 53, 61, 97, 109, 137, 173, 199, 227, 257, 269, 271, 313, 347, 353, 379, 397, 401, 409, 421, 433, 439, 509, 521, 577, 599, 601, 617, 641, 643, 647, 659, 677, 733, 761, 773, 809, 821, 827, 839, 863, 911, 919, 929, 937, 941, 947, 953, 971, 1009
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OFFSET

1,1


COMMENTS

These primes are a subset of the nonharmonic primes A092102. Because these primes are analogous to the irregular primes A000928 that divide the numerators of Bernoulli numbers, they might be called Hirregular primes. The density of these primes is about 0.4  very close to the density of irregular primes.
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 < p1 hold, but k <= (p1)/2. This is because, by symmetry, H(p1n) == H(n) (mod p), so that any eligible k lying between (p+1)/2 and p1 would have a counterpart in the range between 1 and (p1)/2. Furthermore, the minimal k cannot be exactly equal to (p1)/2, because then p would be a Wieferich prime (A001220) and would also divide H(Int(p/4)). Thus k <= (p3)/2, and this inequality is sharp because exact equality holds for p = 29, 37, 3373 (see A072984).  John Blythe Dobson, Apr 09 2015


LINKS

Table of n, a(n) for n=1..53.
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Irregular Prime
Wikipedia, Regular prime


MATHEMATICA

n=2; Table[cnt=0; While[cnt==0, p=Prime[n]; k=1; h=0; While[cnt==0 && k<=(p1)/2, h=h+1/k; If[Mod[Numerator[h], p]==0, cnt++ ]; k++ ]; n++ ]; p, {100}]


CROSSREFS

Cf. A072984 (least k such that prime(n) divides A001008(k)).
Sequence in context: A124110 A153768 A360181 * A134307 A279775 A211191
Adjacent sequences: A092191 A092192 A092193 * A092195 A092196 A092197


KEYWORD

nonn


AUTHOR

T. D. Noe, Feb 24 2004


STATUS

approved



