A330077 - OEIS

%I #17 Dec 02 2019 07:09:45

%S 0,1,1,11,1,197,1,67,19,727,1,24593,1,3039,158,767,1,379873,1,19867,

%T 689,19399,1,3446147,41,38119,217,311809,1,1817969,1,7303,4409,112159,

%U 604,47609581,1,175223,8624,15077683,1,94710023,1,93161,8128,376639,1,960227141

%N a(n) = numerator 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) = a(n)/A330078(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="/A330077/b330077.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) = 11 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 numerator of the sum of their reciprocals, 1/1 + 1/2 + 1/3 = 11/6, is 11.

%t h[n_] := Total@ (1/Flatten[Differences /@ Subsets[Divisors[n], {2}]]); Array[Numerator[h[#]] &, 50]

%Y Cf. A000005, A027750, A330076, A330078 (denominators).

%K nonn,frac

%O 1,4

%A _Amiram Eldar_, Nov 30 2019