A324503 - OEIS

%I #14 Sep 08 2022 08:46:24

%S 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,

%T 15,49,15,31,35,33,208,65,19,63,55,39,21,75,56,43,45,45,65,77,25,49,

%U 64,125,71,95,25,55,49,91,24,105,31,61,175,63,33,99,2416,105

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

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

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

%e 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, ...

%e 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.

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

%o (Magma) [Numerator(&+[d / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]

%o (PARI) a(n) = numerator(sumdiv(n, d, d/numdiv(d))); \\ _Michel Marcus_, Mar 03 2019

%o (Sage) [sum(k/sigma(k, 0) for k in n.divisors()).numerator() for n in (1..80)] # _G. C. Greubel_, Mar 04 2019

%Y Cf. A000005, A324504 (denominators).

%K nonn,frac

%O 1,2

%A _Jaroslav Krizek_, Mar 03 2019