A335189 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 harmonic number.

21, 23, 43, 47, 66, 68, 78, 82, 86, 111, 115, 119, 157, 160, 164, 167, 273, 287, 343, 359, 438, 442, 456, 460, 507, 527, 579, 581, 615, 665, 813, 818, 834, 839, 931, 943, 947, 959, 1082, 1090, 1111, 1119, 1140, 1148, 1248, 1288, 1333, 1340, 1346, 1354, 1360, 1367, 1592, 1640, 1641, 1679, 1807, 1847, 2034, 2067, 2069, 2163, 2190

Offset

1,1

Comments

For more numbers in this list (up to 10^6), see one of the links below by Krattenthaler and Rivoal. The first few numbers L for which v_p(H_L-1) = 2 (rather than 1) for some prime p <= L are 43, 2034 and 2069 with corresponding primes 7, 13 and 7.

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

Table of n, a(n) for n=1..63.

David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.

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.

Carlo Sanna, On the p-adic valuation of harmonic numbers, Journal of Number Theory, 166 (2016), 41-46.

Prog

(PARI) h(n) = sum(i=1, n, 1/i);
is(n) = {forprime(p=1, n, if(valuation((numerator(h(n)-1)), p) > 0, return(1))); return(0)};
for(n=1, 1000, if(is(n)==1, print1(n, ", ")))

Crossrefs

Cf. A007757, A131657, A131658, A268112, A334981.

Sequence in context: A227948 A303718 A270108 * A304269 A302679 A302514

Adjacent sequences: A335186 A335187 A335188 * A335190 A335191 A335192

Keyword

nonn

Author

Petros Hadjicostas, May 26 2020

Status

approved