A382487 The number of divisors of n whose largest prime factor is 3.

0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 5, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 3, 0, 0, 1

Offset

1,6

Comments

The number of 3-smooth divisors of n that are not powers of 2.

The number of terms of A065119 that divide n.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A072078(n) - A001511(n).

a(n) = A001511(n) * A007949(n).

a(n) = 0 if and only if n is in A001651.

a(n) = 1 if and only if n is in A306771.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.

In general, the asymptotic mean of the number prime(k+1)-smooth divisors of n that are not prime(k)-smooth, for k >= 1, is (1/(prime(k+1)-1)) * Product_{i=1..k} (prime(i)/(prime(i)-1)).

Dirichlet g.f.: (zeta(s)/(1-1/2^s))*(1/(1-1/3^s) - 1).

Mathematica

a[n_] := (IntegerExponent[n, 2] + 1) * IntegerExponent[n, 3]; Array[a, 100]

Prog

(PARI) a(n) = (valuation(n, 2) + 1) * valuation(n, 3);

Crossrefs

Cf. A001511, A001651, A065119, A072078, A007949, A051064, A169611, A301461, A306771.

Sequence in context: A033719 A171608 A386604 * A307985 A024164 A394261

Adjacent sequences: A382484 A382485 A382486 * A382488 A382489 A382490

Keyword

nonn,easy

Author

Amiram Eldar, Mar 29 2025

Status

approved