A335189 -id:A335189 - cp's OEIS frontend

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.

43, 2034, 2069, 9702, 9712, 67258, 102691, 102727, 147253, 904332

Offset: 1

Petros Hadjicostas, May 26 2020

This is a subset of A335189. All numbers in this list were copied from one of the links below by Krattenthaler and Rivoal.
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.
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.

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.

Cf. A007757, A131657, A131658, A268112, A335189.

A335207_list := proc(bound) local p, h, H, L, n;
L := NULL; h := 0;
for n from 1 to bound do
    h := h + 1/n; H := h - 1; p:= 2;
    while p <= n do
        if padic:-ordp(H, p) <= 1
           then p := nextprime(p);
           else L := L, n; break;
        fi
    od;
od; L end:
A335207_list(2222); # Peter Luschny, May 29 2020

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

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.

Original entry on oeis.org

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

Petros Hadjicostas, May 26 2020

nonn

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.

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.

Cf. A268112, A335189, A335207.

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; ); }

cp's OEIS Frontend

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.

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