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

Author

T. D. Noe, Feb 20 2004

Keywords

• nonn

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.

Crossrefs

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

Programs

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

Extensions

More terms from Max Alekseyev, May 13 2010