A268112 Numbers k for which the numerator of the k-th harmonic number H_k is divisible by the third power of a prime less than k.

848, 9338, 10583, 3546471722268916272

Offset: 1

Author

Felix Fröhlich, Jan 26 2016

Keywords

• nonn
• hard
• more

Comments

The sequence contains numbers k for which there is a prime p < k with v_p(H_k) >= 3, where v_p(x) is the p-adic valuation of x and H_k is the k-th Harmonic number. All numbers were found by D. W. Boyd. The corresponding p for a(1) through a(4) is 11 while for a(5) (in the b-file) is 83. [Edited by Petros Hadjicostas, May 25 2020]

It is a widely believed conjecture that there is no pair of an integer k and a prime p for which v_p(H_k) >= 4. If variations of this conjecture hold, then Krattenhaler and Rivoal (2007-2009) would be able to establish some formulas for their theory. See also A007757, A131657, and A131658. - Petros Hadjicostas, May 25 2020

Terms a(4) and a(5) are conjectural as possible existence of smaller terms is not eliminated. Carofiglio et al. (2025) computed some further terms (see links). - Max Alekseyev, Apr 01 2025

Links

• Petros Hadjicostas, Table of n, a(n) for n = 1..5
• David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.
• Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.
• L. Carofiglio and G. Cherubini, A copy of the supplementary file with some large terms from Carofiglio et al. (2025) study.
• Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
• Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
• Tamás Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015), 73-94.

Crossrefs

Cf. A001008, A007757, A131657, A131658.

Programs

PARI

h(n) = sum(i=1, n, 1/i);
is(n) = {forprime(p=1, n-1, if(valuation((numerator(h(n))), p) > 2, return(1))); return(0)} \\ Edited by Petros Hadjicostas, May 25 2020

Extensions

Name edited by and a(5) copied from the references by Petros Hadjicostas, May 25 2020