A330078 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.

1, 1, 2, 6, 4, 60, 6, 28, 24, 360, 10, 3960, 12, 1820, 105, 280, 16, 85680, 18, 4560, 630, 13860, 22, 425040, 120, 28600, 234, 98280, 28, 254475, 30, 2480, 5280, 89760, 595, 5654880, 36, 143412, 11115, 2489760, 40, 15595580, 42, 36120, 3465, 318780, 46, 103879776

Offset

1,3

Comments

Erdős and Nicolas conjectured that H(n) = A330077(n)/a(n) < d(n) for all n > 5040.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

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

Example

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.

Mathematica

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

Crossrefs

Cf. A000005, A027750, A330076, A330077 (numerators).

Sequence in context: A215408 A264609 A126262 * A258324 A080499 A072513

Adjacent sequences: A330075 A330076 A330077 * A330079 A330080 A330081

Keyword

nonn,frac

Author

Amiram Eldar, Nov 30 2019

Status

approved