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Harmonic primes.
10

%I #18 Mar 17 2014 04:21:12

%S 5,13,17,23,41,67,73,79,107,113,139,149,157,179,191,193,223,239,241,

%T 251,263,277,281,293,307,311,317,331,337,349,431,443,449,461,467,479,

%U 487,491,499,503,541,547,557,563,569,593,619,653,683,691,709,757,769,787

%N Harmonic primes.

%C 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.

%C 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

%H Charles R Greathouse IV, <a href="/A092101/b092101.txt">Table of n, a(n) for n = 1..10000</a>

%H David W. Boyd, <a href="http://www.emis.de/journals/EM/expmath/volumes/3/3.html">A p-adic study of the partial sums of the harmonic series</a>, Experimental Math., Vol. 3 (1994), No. 4, 287-302.

%H A. Eswarathasan and E. Levine, <a href="http://dx.doi.org/10.1016/0012-365X(90)90234-9">p-integral harmonic sums</a>, Discrete Math. 91 (1991), 249-257.

%o (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

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

%K nonn

%O 1,1

%A _T. D. Noe_, Feb 20 2004

%E More terms from _Max Alekseyev_, May 13 2010