A335210 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.

16, 19, 81, 211, 231, 232, 242, 243, 267, 274, 340, 357, 559, 637, 644, 898, 1121, 1391, 1399, 1412, 1433, 1436, 1439, 1470, 1474, 1501, 1892, 2304, 2336, 2477, 2496, 2520, 2768, 2948, 2992, 3351, 3367, 3563, 3953, 3966, 4431, 4505, 4587, 4596, 4626, 5061, 6058, 6781, 6847, 6861

Offset: 1

Author

Petros Hadjicostas, May 26 2020

Keywords

• nonn

Comments

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.

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.

Links

• 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, Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps", 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.]
• 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.

Crossrefs

Cf. A268112, A335189, A335207.

Programs

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; ); }