A324499 a(n) = numerator of Sum_{d|n} sigma(d)/tau(d) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of the divisors of k (A000005).

1, 5, 3, 29, 4, 15, 5, 103, 22, 10, 7, 29, 8, 25, 12, 887, 10, 55, 11, 58, 15, 35, 13, 103, 43, 20, 52, 145, 16, 30, 17, 1517, 21, 25, 20, 319, 20, 55, 24, 103, 22, 75, 23, 203, 88, 65, 25, 887, 24, 215, 30, 116, 28, 130, 28, 515, 33, 40, 31, 58, 32, 85, 110

Offset

1,2

Comments

Sum_{d|n} sigma(d)/tau(d) > 1 for all n > 1.

Sum_{d|n} sigma(d)/tau(d) = n only for numbers n = 1, 3, 10 and 30.

Links

Table of n, a(n) for n=1..63.

Example

Sum_{d|n} sigma(d)/tau(d) for n >= 1: 1, 5/2, 3, 29/6, 4, 15/2, 5, 103/12, 22/3, 10, 7, 29/2, 8, 25/2, 12, 887/60, ...
For n=4; Sum_{d|4} sigma(d)/tau(d) = sigma(1)/tau(1) + sigma(2)/tau(2) + sigma(4)/tau(4) = 1/1 + 3/2 + 7/3 = 29/6; a(4) = 29.

Mathematica

Table[Numerator[Sum[DivisorSigma[1, k]/DivisorSigma[0, k], {k, Divisors[n]}]], {n, 1, 100}] (* G. C. Greubel, Mar 04 2019 *)

Prog

(Magma) [Numerator(&+[SumOfDivisors(d) / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
(PARI) a(n) = numerator(sumdiv(n, d, sigma(d)/numdiv(d))); \\ Michel Marcus, Mar 03 2019
(Sage) [sum(sigma(k, 1)/sigma(k, 0) for k in n.divisors() ).numerator() for n in (1..100)] # G. C. Greubel, Mar 04 2019

Crossrefs

Cf. A000005, A000203, A323779, A323780, A323781, A324500 (denominators).

Sequence in context: A288812 A288898 A290293 * A189747 A279066 A179210

Adjacent sequences: A324496 A324497 A324498 * A324500 A324501 A324502

Keyword

nonn,frac

Author

Jaroslav Krizek, Mar 02 2019

Status

approved