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The number of prime divisors of n that are not greater than 5, counted with multiplicity.
3

%I #20 Jul 26 2022 10:10:08

%S 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,

%T 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,

%U 1,2,0,5,0,1,3,2,0,2,0,5,4,1,0,3,1,1,1

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

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

%H Amiram Eldar, <a href="/A356006/b356006.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

%o (Python)

%o from sympy import multiplicity as v

%o def a(n): return v(2, n) + v(3, n) + v(5, n)

%o print([a(n) for n in range(1, 88)]) # _Michael S. Branicky_, Jul 25 2022

%Y Cf. A007814, A007949, A112765.

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

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, Jul 23 2022