%I #26 May 12 2024 10:03:34
%S 12,24,60,120,180,240,360,420,720,840,1260,1680,2520,5040
%N Numbers k such that H(k) > d(k), where H(k) = Sum_{1 <= i < j <= d(k)} 1/(d_j - d_i) is sum over ordered pairs of divisors of k, and d(k) is the number of divisors of k.
%C Erdős and Nicolas conjectured that this sequence is finite with only 14 terms.
%C Nicolas (2005) states that the conjecture has been checked up to 10^6.
%C Conjecture checked up to 2*10^10. - _Giovanni Resta_, Dec 01 2019
%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 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="https://doi.org/10.1007/s11139-005-0836-2">Some open questions</a>, The Ramanujan Journal, Vol. 9 (2005), pp. 251-264, <a href="http://math.univ-lyon1.fr/homes-www/nicolas/openquestions.pdf">alternative link</a>.
%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 12 is in the sequence since it has d(12) = 6 divisors, {1, 2, 3, 4, 6, 12}, and the sum of the reciprocals of all the differences between pairs of divisors, {1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 8, 9, 10, 11} is 24593/3960 > 6.
%t Select[Range[5100], Total@ (1 / Flatten[Differences /@ Subsets[(d = Divisors[#]), {2}]]) > Length[d] &]
%Y Cf. A000005, A027750, A330077, A330078.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Nov 30 2019