A372713 - OEIS

%I #26 Jul 19 2024 13:29:17

%S 2,4,3,6,4,6,4,8,4,8,4,9,4,8,6,10,4,8,4,12,6,8,4,12,6,8,5,12,4,12,4,

%T 12,6,8,8,12,4,8,6,16,4,12,4,12,8,8,4,15,6,12,6,12,4,10,8,16,6,8,4,18,

%U 4,8,8,14,8,12,4,12,6,16,4,16,4,8,9,12,8,12,4,20

%N Number of divisors of 3n; a(n) = tau(3*n) = A000005(3*n).

%C In general, for p prime, Sum_{j=1..n} tau(j*p) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

%C If n is in A033428, then a(n) is odd and vice versa. - _R. J. Mathar_, _Amiram Eldar_, May 20 2024.

%H Antti Karttunen, <a href="/A372713/b372713.txt">Table of n, a(n) for n = 1..65537</a>

%F Sum_{k=1..n} a(k) ~ n * (5*(log(n) + 2*gamma - 1) + log(3)) / 3, where gamma is the Euler-Mascheroni constant A001620.

%t Table[DivisorSigma[0, 3*n], {n, 1, 150}]

%o (PARI) a(n) = numdiv(3*n); \\ _Michel Marcus_, May 20 2024

%Y Cf. A000005, A001620, A033428, A099777, A372714, A372715, A372786, A372789, A372792.

%Y Cf. also A144613.

%K nonn

%O 1,1

%A _Vaclav Kotesovec_, May 11 2024