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

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, 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, 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

Offset: 2

Author

T. D. Noe, Feb 24 2004; corrected Jul 28 2004

Keywords

• nonn

Comments

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.

Boyd's table 3 states incorrectly that harmonic primes have 2 generations; harmonic primes have 3 generations.

Examples

			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.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 2..220
• David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302. [WARNING: Table 2 contains miscalculations for p=19, 47, 59, ... - _Max Alekseyev_, Apr 01 2025]
• Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.
• A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.

Crossrefs

Cf. A072984 (least k such that prime(n) divides A001008(k)), A092101 (harmonic primes), A092102 (non-harmonic primes).

Extensions

a(8), a(15), a(17) corrected, and terms a(23) onward from Carofiglio et al. (2025) added by Max Alekseyev, Apr 01 2025