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

Author

Petros Hadjicostas, May 26 2020

Keywords

• nonn
• more
• hard

Comments

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.

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. A007757, A131657, A131658, A268112, A335189.

Programs

Maple

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

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