login
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.
3
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
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
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Nov 30 2019
STATUS
approved