A324503 a(n) = numerator of Sum_{d|n} (d/tau(d)) where tau(k) = the number of divisors of k (A000005).

1, 2, 5, 10, 7, 5, 9, 16, 11, 7, 13, 25, 15, 9, 35, 128, 19, 11, 21, 35, 45, 13, 25, 40, 71, 15, 49, 15, 31, 35, 33, 208, 65, 19, 63, 55, 39, 21, 75, 56, 43, 45, 45, 65, 77, 25, 49, 64, 125, 71, 95, 25, 55, 49, 91, 24, 105, 31, 61, 175, 63, 33, 99, 2416, 105

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..65.

Formula

a(p) = p + 2 for p = odd primes.

Example

Sum_{d|n} (d/tau(d)) for n >= 1: 1, 2, 5/2, 10/3, 7/2, 5, 9/2, 16/3, 11/2, 7, 13/2, 25/3, 15/2, 9, 35/4, 128/15, ...
For n=4; Sum_{d|4} (d/tau(d)) = 1/tau(1) + 2/tau(2) + 4/tau(4) = 1/1 + 2/2 + 4/3 = 10/3; a(4) = 10.

Mathematica

Table[Numerator[Sum[k/DivisorSigma[0, k], {k, Divisors[n]}]], {n, 1, 80}] (* G. C. Greubel, Mar 04 2019 *)

Prog

(Magma) [Numerator(&+[d / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
(PARI) a(n) = numerator(sumdiv(n, d, d/numdiv(d))); \\ Michel Marcus, Mar 03 2019
(Sage) [sum(k/sigma(k, 0) for k in n.divisors()).numerator() for n in (1..80)] # G. C. Greubel, Mar 04 2019

Crossrefs

Cf. A000005, A324504 (denominators).

Sequence in context: A175467 A361806 A188525 * A126842 A291933 A332198

Adjacent sequences: A324500 A324501 A324502 * A324504 A324505 A324506

Keyword

nonn,frac

Author

Jaroslav Krizek, Mar 03 2019

Status

approved