A330076 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.

12, 24, 60, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040

Offset

1,1

Comments

Erdős and Nicolas conjectured that this sequence is finite with only 14 terms.

Nicolas (2005) states that the conjecture has been checked up to 10^6.

Conjecture checked up to 2*10^10. - Giovanni Resta, Dec 01 2019

References

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.

Links

Table of n, a(n) for n=1..14.

Paul Erdős and Jean-Louis Nicolas, On functions connected with prime divisors of an integer, 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.

Jean-Louis Nicolas, Some open questions, The Ramanujan Journal, Vol. 9 (2005), pp. 251-264, alternative link.

Gérald Tenenbaum, Une inégalité de Hilbert pour les diviseurs, Indagationes Mathematicae, Vol. 2, No. 1 (1991), pp. 105-114.

Example

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.

Mathematica

Select[Range[5100], Total@ (1 / Flatten[Differences /@ Subsets[(d = Divisors[#]), {2}]]) > Length[d] &]

Crossrefs

Cf. A000005, A027750, A330077, A330078.

Sequence in context: A230355 A063975 A227895 * A332037 A332039 A001335

Adjacent sequences: A330073 A330074 A330075 * A330077 A330078 A330079

Keyword

nonn,more

Author

Amiram Eldar, Nov 30 2019

Status

approved