cp's OEIS Frontend

A092103 Number of values of k for which prime(n) divides A001008(k), the numerator of the k-th harmonic number.

%I A092103 #67 Apr 03 2025 04:44:26
%S A092103 3,3,13,638,3,3,25,3,18,26,15,3,27,24,17,23,13,3,45,3,3,43038,7,74,44,
%T A092103 63,3,1273,3,3515,7,38,3,3,7,3,74,526,288,3,19,3,3,41,11,59,3,31,65,
%U A092103 176,3,3,3,20,3,106,55,3,3,89,3,3,3,79,3,3,3,47,3,21,253,29,7,79,41,19,701533,13,9,703,23,3,205,105,3,3,323,3,7,3,3,3,3,3,3,13,1763
%N A092103 Number of values of k for which prime(n) divides A001008(k), the numerator of the k-th harmonic number.
%C A092103 For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k). This sequence gives the size of J_p. A072984 and A177734 give the smallest and largest elements of J_p, respectively.
%C A092103 A092101 gives primes prime(n) such that a(n) = 3 (i.e., a(A000720(A092101(m))) = 3 for all m). A092102 gives primes prime(n) such that a(n) > 3.
%C A092103 From _Carlo Sanna_, Apr 06 2016: (Start)
%C A092103 Eswarathasan and Levine conjectured that for any prime number p the set J_p is finite.
%C A092103 I proved that if J_p(x) is the number of integers in J_p that are less than x > 1, then J_p(x) < 129 p^(2/3) x^0.765 for any prime p. In particular, J_p has asymptotic density zero. (End)
%C A092103 Bing-Ling Wu and Yong-Gao Chen improved Sanna's (see previous comment) result showing that J_p(x) <= 3 x^(2/3 + 1/(25 log p)) for any prime p and any x > 1. - _Carlo Sanna_, Jan 12 2017
%H A092103 Max Alekseyev, <a href="/A092103/b092103.txt">Table of n, a(n) for n = 2..220</a>
%H A092103 David W. Boyd, <a href="http://projecteuclid.org/euclid.em/1048515811">A p-adic study of the partial sums of the harmonic series</a>, Experimental Math., Vol. 3 (1994), No. 4, 287-302. [WARNING: Table 2 contains miscalculations for p=19, 47, 59, ... - _Max Alekseyev_, Oct 23 2012]
%H A092103 Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, <a href="https://arxiv.org/abs/2503.15714">On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers</a>, arXiv:2503.15714 [math.NT], 2025.
%H A092103 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.
%H A092103 Carlo Sanna, <a href="http://dx.doi.org/10.1016/j.jnt.2016.02.020">On the p-adic valuation of harmonic numbers</a>, J. Number Theory 166 (2016), 41-46.
%H A092103 Bing-Ling Wu and Yong-Gao Chen, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.027">On certain properties of harmonic numbers</a>, J. Number Theory 175 (2017), 66-86.
%e A092103 a(2) = 3 because 3 divides A001008(k) for k = 2, 7, and 22.
%e A092103 a(4) = 13 because 7 divides A001008(k) for only the 13 values k = 6, 42, 48, 295, 299, 337, 341, 2096, 2390, 14675, 16731, 16735, and 102728. This is the 4th row in A229493.
%Y A092103 Cf. A072984, A092101, A092102.
%Y A092103 Cf. A092193 (number of generations for each prime).
%Y A092103 Cf. A229493 (terms for each prime).
%K A092103 nonn
%O A092103 2,1
%A A092103 _T. D. Noe_, Feb 20 2004
%E A092103 a(8), a(15), and a(17) corrected by _Max Alekseyev_, Oct 23 2012
%E A092103 Terms a(23) onward from Carofiglio et al. (2025) added by _Max Alekseyev_, Apr 01 2025