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A335207 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

%I A335207 #11 May 30 2020 09:47:42
%S A335207 43,2034,2069,9702,9712,67258,102691,102727,147253,904332
%N A335207 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.
%C A335207 This is a subset of A335189. All numbers in this list were copied from one of the links below by Krattenthaler and Rivoal.
%C A335207 For all L in this list (up to 904332), we have v_p(H_L - 1) = 2 with corresponding primes as follows: p(1) = 7, p(2) = 13, p(3) = 7, p(4) = p(5) = 11, p(6) = 41, p(7) = p(8) = 11, p(9) = 53, and p(10) = 97.
%C A335207 The calculation of v_p(H_L-1) and v_p(H_L) for all primes p <= L is related to some results about the integrality of the Taylor coefficients of mirror maps. See Theorems 3 and 4 in Krattenthaler and Rivoal (2007-2009, 2009) and sequences A007757, A131657, and A131658.
%H A335207 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 A335207 Christian Krattenthaler and Tanguy Rivoal, <a href="https://www.mat.univie.ac.at/~kratt/artikel/H1.html">Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps"</a>, 2007-2009. [This table contains all triplets of numbers (L, p, v_p(H_L - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H_L - 1) > 0.]
%H A335207 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.
%p A335207 A335207_list := proc(bound) local p, h, H, L, n;
%p A335207 L := NULL; h := 0;
%p A335207 for n from 1 to bound do
%p A335207     h := h + 1/n; H := h - 1; p:= 2;
%p A335207     while p <= n do
%p A335207         if padic:-ordp(H, p) <= 1
%p A335207            then p := nextprime(p);
%p A335207            else L := L, n; break;
%p A335207         fi
%p A335207     od;
%p A335207 od; L end:
%p A335207 A335207_list(2222); # _Peter Luschny_, May 29 2020
%o A335207 (PARI) list(nn) = {my(h=-1); for (n=1, nn, h += 1/n; forprime(p=1, n-1, if(valuation(h, p) > 1, print1(n, ", "); break)););} \\ _Petros Hadjicostas_, May 26 2020, courtesy of _Michel Marcus_
%Y A335207 Cf. A007757, A131657, A131658, A268112, A335189.
%K A335207 nonn,more,hard
%O A335207 1,1
%A A335207 _Petros Hadjicostas_, May 26 2020