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%I #18 Dec 02 2019 04:11:32
%S 1,1,2,6,4,60,6,28,24,360,10,3960,12,1820,105,280,16,85680,18,4560,
%T 630,13860,22,425040,120,28600,234,98280,28,254475,30,2480,5280,89760,
%U 595,5654880,36,143412,11115,2489760,40,15595580,42,36120,3465,318780,46,103879776
%N a(n) = denominator of Sum_{1 <= i < j <= d(n)} 1/(d_j - d_i), sum over ordered pairs of divisors of n, where d(n) is the number of divisors of n.
%C Erdős and Nicolas conjectured that H(n) = A330077(n)/a(n) < d(n) for all n > 5040.
%D Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS 84, American Mathematical Society, 1994, problem 23. p. 200.
%H Amiram Eldar, <a href="/A330078/b330078.txt">Table of n, a(n) for n = 1..10000</a>
%H Paul Erdős and Jean-Louis Nicolas, <a href="https://users.renyi.hu/~p_erdos/1989-17.pdf">On functions connected with prime divisors of an integer</a>, Number Theory and Applications, Proceedings of the NATO Advanced Study Institute, Banff Centre, Canada, April 27-May 5, 1988, (R. A. Mollin, ed.), Kluwer Academic Publishers, 1989, pp. 381-391.
%H Jean-Louis Nicolas, <a href="http://math.univ-lyon1.fr/homes-www/nicolas/openquestions.pdf">Some open questions</a>, The Ramanujan Journal, Vol. 9 (2005), pp. 251-264.
%H Gérald Tenenbaum, <a href="https://doi.org/10.1016/0019-3577(91)90046-A">Une inégalité de Hilbert pour les diviseurs</a>, Indagationes Mathematicae, Vol. 2, No. 1 (1991), pp. 105-114.
%e a(4) = 6 since the divisors of 4 are {1, 2, 4}, the differences between ordered pairs of divisors are 2-1 = 1, 4-2 = 2, and 4-1 = 3, and the denominator of the sum of their reciprocals, 1/1 + 1/2 + 1/3 = 11/6, is 6.
%t h[n_] := Total@ (1/Flatten[Differences /@ Subsets[Divisors[n], {2}]]); Array[Denominator[h[#]] &, 50]
%Y Cf. A000005, A027750, A330076, A330077 (numerators).
%K nonn,frac
%O 1,3
%A _Amiram Eldar_, Nov 30 2019
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