A092101 Harmonic primes.

5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349, 431, 443, 449, 461, 467, 479, 487, 491, 499, 503, 541, 547, 557, 563, 569, 593, 619, 653, 683, 691, 709, 757, 769, 787

Offset

1,1

Comments

For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k), the numerator of the harmonic number H(k). For harmonic primes, J_p contains only the three numbers p-1, (p-1)p and (p-1)(p+1). It has been conjectured that there are an infinite number of these primes and that their density in the primes is 1/e.

Prime p=A000040(n) is in this sequence iff neither H(k) == 0 (mod p), nor H(k) == -A177783(n) (mod p) have solutions for 1 <= k <= p-2. - Max Alekseyev, May 13 2010

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302.

A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.

Prog

(PARI) is(p)=my(K=-Mod((binomial(2*p-1, p)-1)/2/p^3, p), H=Mod(0, p)); for(k=1, p-2, H+=1/k; if(H==0||H==K, return(0))); 1 \\ Charles R Greathouse IV, Mar 16 2014

Crossrefs

Cf. A092102 (non-harmonic primes), A092103 (size of J_p).

Sequence in context: A182078 A074278 A087895 * A105596 A037046 A126887

Adjacent sequences: A092098 A092099 A092100 * A092102 A092103 A092104

Keyword

nonn

Author

T. D. Noe, Feb 20 2004

Extensions

More terms from Max Alekseyev, May 13 2010

Status

approved