A335210 - OEIS

%I #13 May 27 2020 16:27:40

%S 16,19,81,211,231,232,242,243,267,274,340,357,559,637,644,898,1121,

%T 1391,1399,1412,1433,1436,1439,1470,1474,1501,1892,2304,2336,2477,

%U 2496,2520,2768,2948,2992,3351,3367,3563,3953,3966,4431,4505,4587,4596,4626,5061,6058,6781,6847,6861

%N Numbers L such that there is a prime p <= L for which v_p(H'(L) - 1) > 0, where v_p(x) is the p-adic valuation of x and H'(L) is the L-th alternating harmonic number.

%C This sequence was inspired by the database of Krattenthaler and Rivoal (see the link below) about 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. Here v_p(x) is the p-adic valuation of x and H(L) is the L-th harmonic number. See also the sequences A268112, A335189, and A335207.

%C Here we tabulate the numbers L >= 1 for which there is a prime p <= L such that v_p(H'(L) - 1) >= 1, where H'(L) = Sum_{k=1..L} (-1)^(k+1)/k. The first few numbers L for which v_p(H'(L) - 1) = 2 (rather than 1) for some p <= L are 1501, 4596, and 9367 with corresponding p equal to 7, 19, and 37, respectively.

%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="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 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.

%o (PARI) listaa(nn) = {my(h=0, s=1, nh); for (n=1, nn, h += s/n; nh = numerator(h-1); forprime(p=1, n-1, if(valuation(nh, p) > 0, print1(n, ", "); break)); s = -s; ); }

%Y Cf. A268112, A335189, A335207.

%K nonn

%O 1,1

%A _Petros Hadjicostas_, May 26 2020