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%I #57 May 29 2020 04:27:35
%S 30,241,1057,1499,10499,318425
%N 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.
%C 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.
%C 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.
%C 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.
%C 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.
%C 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.
%C It holds v_7(H'(10499)) = 2 and v_691(H'(318425)) = 2. a(7) > 5*10^5. - _Giovanni Resta_, May 26 2020
%H David W. Boyd, <a href="https://doi.org/10.1080/10586458.1994.10504298">A p-adic study of the partial sum of the harmonic series</a>, Experimental Mathematics, 3(4) (1994), 287-302.
%H Christian Krattenthaler and Tanguy Rivoal, <a href="http://arxiv.org/abs/0709.1432">On the integrality of the Taylor coefficients of mirror maps</a>, arXiv:0709.1432 [math.NT], 2007-2009.
%H Christian Krattenthaler and Tanguy Rivoal, <a href="http://dx.doi.org/10.4310/CNTP.2009.v3.n3.a5">On the integrality of the Taylor coefficients of mirror maps, II</a>, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
%H Tamás Lengyel, <a href="http://dx.doi.org/10.1016/j.jnt.2014.09.015">On p-adic properties of the Stirling numbers of the first kind</a>, Journal of Number Theory, 148 (2015), 73-94.
%o (PARI) ah(n) = sum(i=1, n, (-1)^(i+1)/i);
%o is(n) = {forprime(p=1, n-1, if(valuation((numerator(ah(n))), p) > 1, return(1))); return(0)}
%o (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
%Y Cf. A007757, A131657, A131658, A268112 (similar sequence for harmonic numbers).
%K nonn,hard,more
%O 1,1
%A _Petros Hadjicostas_, May 25 2020
%E a(5) from _Michel Marcus_, May 26 2020
%E a(6) from _Giovanni Resta_, May 26 2020
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