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 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. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 10 03:15 EDT 2023. Contains 363186 sequences. (Running on oeis4.)