OFFSET
1,1
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
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.
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.
PROG
(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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Felix Fröhlich, Jan 26 2016
EXTENSIONS
Name edited by and a(5) copied from the references by Petros Hadjicostas, May 25 2020
STATUS
approved