A372713 Number of divisors of 3n; a(n) = tau(3*n) = A000005(3*n).

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

Offset: 1

Author

Vaclav Kotesovec, May 11 2024

Keywords

• nonn

Comments

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.

If n is in A033428, then a(n) is odd and vice versa. - R. J. Mathar, Amiram Eldar, May 20 2024.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

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

Cf. also A144613.

Programs

Mathematica

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

PARI

a(n) = numdiv(3*n); \\ Michel Marcus, May 20 2024

Formula

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