A355584 - OEIS

%I #17 Dec 25 2022 02:11:17

%S 1,3,4,7,6,12,1,15,13,18,1,28,1,3,24,31,1,39,1,42,4,3,1,60,31,3,40,7,

%T 1,72,1,63,4,3,6,91,1,3,4,90,1,12,1,7,78,3,1,124,1,93,4,7,1,120,6,15,

%U 4,3,1,168,1,3,13,127,6,12,1,7,4,18,1,195,1,3,124,7

%N a(n) is the sum of the 5-smooth divisors of n.

%H Amiram Eldar, <a href="/A355584/b355584.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 5, and 1 otherwise.

%F a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)*(5^(A112765(n)+1)-1)/8.

%F a(n) = A000203(A355582(n)).

%F a(n) <= A000203(n), with equality if and only if n is in A051037.

%F Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3))*(5^s/(5^s-5)). - _Amiram Eldar_, Dec 25 2022

%t a[n_] := (Times @@ ({2, 3, 5}^(IntegerExponent[n, {2, 3, 5}] + 1) - 1))/8; Array[a, 100]

%o (PARI) a(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;

%o (Python)

%o from sympy import multiplicity as v

%o def a(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1) * (5**(v(5, n)+1)-1) // 8

%o print([a(n) for n in range(1, 77)]) # _Michael S. Branicky_, Jul 08 2022

%Y Sum of the p-smooth divisors of n: A038712 (2), A072079 (3), this sequence (5).

%Y Cf. A000203, A007814, A007949, A051037, A112765, A355582, A355583.

%K nonn,mult,easy

%O 1,2

%A _Amiram Eldar_, Jul 08 2022