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

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, 63, 3, 1273, 3, 3515, 7, 38, 3, 3, 7, 3, 74, 526, 288, 3, 19, 3, 3, 41, 11, 59, 3, 31, 65, 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

Offset

2,1

Comments

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.

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.

From Carlo Sanna, Apr 06 2016: (Start)

Eswarathasan and Levine conjectured that for any prime number p the set J_p is finite.

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)

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

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, Oct 23 2012]

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.

Carlo Sanna, On the p-adic valuation of harmonic numbers, J. Number Theory 166 (2016), 41-46.

Bing-Ling Wu and Yong-Gao Chen, On certain properties of harmonic numbers, J. Number Theory 175 (2017), 66-86.

Example

a(2) = 3 because 3 divides A001008(k) for k = 2, 7, and 22.
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.

Crossrefs

Cf. A072984, A092101, A092102.

Cf. A092193 (number of generations for each prime).

Cf. A229493 (terms for each prime).

Sequence in context: A019154 A255675 A272857 * A233697 A284041 A288294

Adjacent sequences: A092100 A092101 A092102 * A092104 A092105 A092106

Keyword

nonn

Author

T. D. Noe, Feb 20 2004

Extensions

a(8), a(15), and a(17) corrected by Max Alekseyev, Oct 23 2012

Terms a(23) onward from Carofiglio et al. (2025) added by Max Alekseyev, Apr 01 2025

Status

approved