OFFSET
1,1
COMMENTS
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.
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
LINKS
Robert Israel, Table of n, a(n) for n = 1..2500
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Irregular Prime
Wikipedia, Regular prime
MAPLE
A1008:= 1: h:= 1: for n from 2 to 10000 do h:= h+1/n; A1008:= A1008, numer(h) od:
A1008:= [A1008]:
filter:= proc(p) ormap(i -> A1008[i] mod p = 0, [$1..p-2]) end proc:
select(filter, [seq(ithprime(i), i=1..1000)]); # Robert Israel, Jan 20 2025
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
T. D. Noe, Feb 24 2004
STATUS
approved