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

1,4

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

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A007814, A007949, A112765.

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

Sequence in context: A177995 A332104 A238735 * A258120 A147786 A275019

Adjacent sequences: A356003 A356004 A356005 * A356007 A356008 A356009

Keyword

nonn,easy

Author

Amiram Eldar, Jul 23 2022

Status

approved