

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

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.


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



KEYWORD

nonn,frac


AUTHOR



STATUS

approved



