

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
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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 JeanLouis 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 27May 5, 1988, (R. A. Mollin, ed.), Kluwer Academic Publishers, 1989, pp. 381391.
JeanLouis Nicolas, Some open questions, The Ramanujan Journal, Vol. 9 (2005), pp. 251264.
Gérald Tenenbaum, Une inégalité de Hilbert pour les diviseurs, Indagationes Mathematicae, Vol. 2, No. 1 (1991), pp. 105114.


EXAMPLE

a(4) = 6 since the divisors of 4 are {1, 2, 4}, the differences between ordered pairs of divisors are 21 = 1, 42 = 2, and 41 = 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



