

A334981


Numbers k such that the numerator of the kth alternating harmonic number H'(k) is divisible by the square of a prime less than k.


1




OFFSET

1,1


COMMENTS

This sequence was inspired by unsolved conjectures related to the papers by Boyd (1994) and Krattenhaler and Rivoal (20072009, 2009) about the harmonic numbers H(k) = Sum_{i=1..k} 1/i. See also the comments for sequences A007757, A131657, A131658, and A268112. Here we are dealing with the alternating harmonic numbers H'(k) = Sum_{i=1..k} (1)^(i+1)/i.
For the harmonic numbers H(k), it is not known whether there is k >= 1 and a prime p such that v_p(H(k)) >= 4, where v_p(x) is the padic valuation of x. Since p cannot be present in both the numerator and the denominator of H(k), this is equivalent to saying that the numerator of H(k) cannot be divisible by the fourth power of a prime p.
If variations of the above conjecture are true, then some conditional results in Krattenhaler and Rivoal (20072009, 2009) would hold. Boyd (1994) found only 5 integers k such that there is a prime p < k with v_p(H(k)) >= 3. Since 1994 no other k's have been found that satisfy the latter inequality.
We claim that a similar conjecture holds for the alternating harmonic numbers H'(k): there is no pair of an integer k and a prime p such that v_p(H'(k)) >= 4; i.e., there is no k for which the numerator of H'(k) is divisible by the fourth power of a prime.
This sequence contains those k's for which there is a prime p < k with v_p(H'(k)) >= 2. Up to 2000, we have only been able to find four such k's. The corresponding primes for 30, 241, 1057, and 1499 are 7, 19, 37, and 7. We have v_7(H'(30)) = v_19(H'(241)) = v_37(H'(1057)) = 2, while v_7(H'(1499)) = 3.
It holds v_7(H'(10499)) = 2 and v_691(H'(318425)) = 2. a(7) > 5*10^5.  Giovanni Resta, May 26 2020


LINKS

Table of n, a(n) for n=1..6.
David W. Boyd, A padic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287302.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 20072009.
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), 555591.
Tamás Lengyel, On padic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015), 7394.


PROG

(PARI) ah(n) = sum(i=1, n, (1)^(i+1)/i);
is(n) = {forprime(p=1, n1, if(valuation((numerator(ah(n))), p) > 1, return(1))); return(0)}
(PARI) listaa(nn) = {my(h=0, s=1, nh); for (n=1, nn, h += s/n; nh = numerator(h); forprime(p=1, n1, if(valuation(nh, p) > 1, print1(n, ", "); break)); s = s; ); } \\ Michel Marcus, May 26 2020


CROSSREFS

Cf. A007757, A131657, A131658, A268112 (similar sequence for harmonic numbers).
Sequence in context: A138404 A136381 A042752 * A230703 A024448 A125367
Adjacent sequences: A334978 A334979 A334980 * A334982 A334983 A334984


KEYWORD

nonn,hard,more


AUTHOR

Petros Hadjicostas, May 25 2020


EXTENSIONS

a(5) from Michel Marcus, May 26 2020
a(6) from Giovanni Resta, May 26 2020


STATUS

approved



