A355583 a(n) is the number of the 5-smooth divisors of n.

1, 2, 2, 3, 2, 4, 1, 4, 3, 4, 1, 6, 1, 2, 4, 5, 1, 6, 1, 6, 2, 2, 1, 8, 3, 2, 4, 3, 1, 8, 1, 6, 2, 2, 2, 9, 1, 2, 2, 8, 1, 4, 1, 3, 6, 2, 1, 10, 1, 6, 2, 3, 1, 8, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 6, 3, 1, 4, 1, 10, 5, 2, 1, 6, 2, 2

Offset

1,2

Links

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

Formula

Multiplicative with a(p^e) = e+1 if p <= 5 and 1 otherwise.

a(n) = (A007814(n) + 1)*(A007949(n) + 1)*(A112765(n) + 1).

a(n) = A000005(A355582(n)).

a(n) <= A000005(n), with equality if and only if n is in A051037.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/4.

Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)*(1-1/5^s)). - Amiram Eldar, Dec 25 2022

Mathematica

a[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

Prog

(PARI) a(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
(Python)
from sympy import multiplicity as v
def a(n): return (v(2, n)+1)*(v(3, n)+1)*(v(5, n)+1)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Jul 08 2022

Crossrefs

Cf. A000005, A007814, A007949, A051037, A072078, A112765, A355582, A355584.

Sequence in context: A340224 A083901 A274517 * A368543 A359302 A366147

Adjacent sequences: A355580 A355581 A355582 * A355584 A355585 A355586

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jul 08 2022

Status

approved