A334981 Numbers k such that the numerator of the k-th alternating harmonic number H'(k) is divisible by the square of a prime less than k.

30, 241, 1057, 1499, 10499, 318425

Offset: 1

Author

Petros Hadjicostas, May 25 2020

Keywords

• nonn
• hard
• more

Comments

This sequence was inspired by unsolved conjectures related to the papers by Boyd (1994) and Krattenhaler and Rivoal (2007-2009, 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 p-adic 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 (2007-2009, 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

• 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, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
• Tamás Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015), 73-94.

Crossrefs

Cf. A007757, A131657, A131658, A268112 (similar sequence for harmonic numbers).

Programs

PARI

ah(n) = sum(i=1, n, (-1)^(i+1)/i);
is(n) = {forprime(p=1, n-1, 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, n-1, if(valuation(nh, p) > 1, print1(n, ", "); break)); s = -s;);} \\ Michel Marcus, May 26 2020

Extensions

a(5) from Michel Marcus, May 26 2020

a(6) from Giovanni Resta, May 26 2020