A324982 a(n) = numerator of Sum_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).

1, 2, 5, 14, 7, 25, 9, 62, 23, 59, 13, 1819, 15, 109, 245, 3382, 19, 1987, 21, 2731, 465, 257, 25, 250747, 271, 355, 775, 22295, 31, 405385, 33, 28434, 1121, 599, 1253, 6726169, 39, 745, 1557, 642763, 43, 1556549, 45, 28657, 61031, 1085, 49, 765671783, 713

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..49.

Formula

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

Example

Sum_{d|n} (pod(d)/tau(d)) for n >= 1: 1, 2, 5/2, 14/3, 7/2, 25/2, 9/2, 62/3, 23/2, 59/2, ...
For n=4; Sum_{d|4} (pod(d)/tau(d)) = pod(1)/tau(1) + pod(2)/tau(2) + pod(4)/tau(4) = 1/1 + 2/2 + 8/3 = 14/3;  a(4) = 14.

Mathematica

Array[Numerator@ DivisorSum[#, Apply[Times, Divisors@ #]/DivisorSigma[0, #] &] &, 49] (* Michael De Vlieger, Mar 24 2019 *)

Prog

(Magma) [Numerator(&+[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
(PARI) a(n) = numerator(sumdiv(n, d, vecprod(divisors(d))/numdiv(d))); \\ Michel Marcus, Mar 23 2019

Crossrefs

Cf. A000203, A007955, A324983 (denominators).

Sequence in context: A279253 A279958 A348881 * A289682 A151854 A146526

Adjacent sequences: A324979 A324980 A324981 * A324983 A324984 A324985

Keyword

nonn,frac

Author

Jaroslav Krizek, Mar 22 2019

Status

approved