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A092193 Number of generations for which prime(n) divides A001008(k) for some k.

%I A092193 #18 Apr 04 2025 09:44:39
%S A092193 4,3,7,30,3,3,8,3,5,7,4,3,5,7,6,7,4,3,8,3,3,339,4,11,10,14,3,47,3,146,
%T A092193 4,8,3,3,4,3,20,49,33,3,6,3,3,11,5,12,3,6,17,21,3,3,3,5,3,20,18,3,3,
%U A092193 14,3,3,3,11,3,3,3,10,3,6,35,8,4,13,11,8,1815,5,4,52,5,3,30,11,3,3,36,3,4,3,3,3,3,3,3,4,61,4,3,3,3,3,3,8,28,4,3,6,4,6,21,19,3,94
%N A092193 Number of generations for which prime(n) divides A001008(k) for some k.
%C A092193 For any prime p, generation m consists of the numbers p^(m-1) <= k < p^m. The zeroth generation consists of just the number 0. When there is a k in generation m such that p divides A001008(k), then that k may generate solutions in generation m+1. It is conjectured that for all primes there are solutions for only a finite number of generations.
%C A092193 Boyd's table 3 states incorrectly that harmonic primes have 2 generations; harmonic primes have 3 generations.
%H A092193 Max Alekseyev, <a href="/A092193/b092193.txt">Table of n, a(n) for n = 2..220</a>
%H A092193 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_, Apr 01 2025]
%H A092193 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 A092193 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.
%e A092193 a(4)=7 because the fourth prime, 7, divides A001008(k) for k = 6, 42, 48, 295, 299, 337, 341, 2096, 2390, 14675, 16731, 16735 and 102728. These values of k fall into 6 generations; adding the zeroth generation makes a total of 7 generations.
%Y A092193 Cf. A072984 (least k such that prime(n) divides A001008(k)), A092101 (harmonic primes), A092102 (non-harmonic primes).
%K A092193 nonn
%O A092193 2,1
%A A092193 _T. D. Noe_, Feb 24 2004; corrected Jul 28 2004
%E A092193 a(8), a(15), a(17) corrected, and terms a(23) onward from Carofiglio et al. (2025) added by _Max Alekseyev_, Apr 01 2025