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

%I A355583 #15 Dec 25 2022 02:11:21
%S A355583 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,
%T A355583 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,
%U A355583 3,2,4,1,12,1,2,6,3,1,4,1,10,5,2,1,6,2,2
%N A355583 a(n) is the number of the 5-smooth divisors of n.
%H A355583 Amiram Eldar, <a href="/A355583/b355583.txt">Table of n, a(n) for n = 1..10000</a>
%F A355583 Multiplicative with a(p^e) = e+1 if p <= 5 and 1 otherwise.
%F A355583 a(n) = (A007814(n) + 1)*(A007949(n) + 1)*(A112765(n) + 1).
%F A355583 a(n) = A000005(A355582(n)).
%F A355583 a(n) <= A000005(n), with equality if and only if n is in A051037.
%F A355583 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/4.
%F A355583 Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)*(1-1/5^s)). - _Amiram Eldar_, Dec 25 2022
%t A355583 a[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Array[a, 100]
%o A355583 (PARI) a(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
%o A355583 (Python)
%o A355583 from sympy import multiplicity as v
%o A355583 def a(n): return (v(2, n)+1)*(v(3, n)+1)*(v(5, n)+1)
%o A355583 print([a(n) for n in range(1, 87)]) # _Michael S. Branicky_, Jul 08 2022
%Y A355583 Cf. A000005, A007814, A007949, A051037, A072078, A112765, A355582, A355584.
%K A355583 nonn,mult,easy
%O A355583 1,2
%A A355583 _Amiram Eldar_, Jul 08 2022