A330077 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.

0, 1, 1, 11, 1, 197, 1, 67, 19, 727, 1, 24593, 1, 3039, 158, 767, 1, 379873, 1, 19867, 689, 19399, 1, 3446147, 41, 38119, 217, 311809, 1, 1817969, 1, 7303, 4409, 112159, 604, 47609581, 1, 175223, 8624, 15077683, 1, 94710023, 1, 93161, 8128, 376639, 1, 960227141

Offset

1,4

Comments

Erdős and Nicolas conjectured that H(n) = a(n)/A330078(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) = 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.

Mathematica

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

Crossrefs

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

Sequence in context: A218018 A093158 A335157 * A132098 A223513 A160480

Adjacent sequences: A330074 A330075 A330076 * A330078 A330079 A330080

Keyword

nonn,frac

Author

Amiram Eldar, Nov 30 2019

Status

approved