|
%I #48 May 29 2020 04:27:43
%S 848,9338,10583,3546471722268916272
%N 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.
%C 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]
%C 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
%H Petros Hadjicostas, <a href="/A268112/b268112.txt">Table of n, a(n) for n = 1..5</a>
%H David W. Boyd, <a href="https://doi.org/10.1080/10586458.1994.10504298">A p-adic study of the partial sum of the harmonic series</a>, Experimental Mathematics, 3(4) (1994), 287-302.
%H Christian Krattenthaler and Tanguy Rivoal, <a href="http://arxiv.org/abs/0709.1432">On the integrality of the Taylor coefficients of mirror maps</a>, arXiv:0709.1432 [math.NT], 2007-2009.
%H Christian Krattenthaler and Tanguy Rivoal, <a href="http://dx.doi.org/10.4310/CNTP.2009.v3.n3.a5">On the integrality of the Taylor coefficients of mirror maps, II</a>, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
%H Tamás Lengyel, <a href="http://dx.doi.org/10.1016/j.jnt.2014.09.015">On p-adic properties of the Stirling numbers of the first kind</a>, Journal of Number Theory, 148 (2015), 73-94.
%o (PARI) h(n) = sum(i=1, n, 1/i);
%o 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
%Y Cf. A001008, A007757, A131657, A131658.
%K nonn,hard,more
%O 1,1
%A _Felix Fröhlich_, Jan 26 2016
%E Name edited by and a(5) copied from the references by _Petros Hadjicostas_, May 25 2020
| 