

A268112


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


5




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 padic valuation of x and H_k is the kth 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 bfile) 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 (20072009) 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 padic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287302.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 20072009.
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), 555591.
Tamás Lengyel, On padic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015), 7394.


PROG

(PARI) h(n) = sum(i=1, n, 1/i);
is(n) = {forprime(p=1, n1, if(valuation((numerator(h(n))), p) > 2, return(1))); return(0)} \\ Edited by Petros Hadjicostas, May 25 2020


CROSSREFS

Cf. A001008, A007757, A131657, A131658.
Sequence in context: A252253 A247532 A235065 * A194618 A252546 A251257
Adjacent sequences: A268109 A268110 A268111 * A268113 A268114 A268115


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



