A356006 The number of prime divisors of n that are not greater than 5, counted with multiplicity.

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

Offset: 1

Author

Amiram Eldar, Jul 23 2022

Keywords

• nonn
• easy

Comments

Equivalently, the number of prime divisors, counted with multiplicity, of the largest 5-smooth divisor of n.

Links

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

Crossrefs

Cf. A007814, A007949, A112765.

Cf. A001222, A112759, A169611, A355582, A355583, A355584.

Programs

Mathematica

a[n_] := Plus @@ IntegerExponent[n, {2, 3, 5}]; Array[a, 100]

PARI

a(n) = valuation(n, 2) + valuation(n, 3) + valuation(n, 5);

Python

from sympy import multiplicity as v
def a(n): return v(2, n) + v(3, n) + v(5, n)
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 25 2022

Formula

Totally additive with a(p) = 1 if p <= 5, and 0 otherwise.

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

a(n) = A001222(A355582(n)).

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7/4.