A355584 a(n) is the sum of the 5-smooth divisors of n.

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

Offset

1,2

Links

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

Formula

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

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

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

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

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

Mathematica

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

Prog

(PARI) a(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;
(Python)
from sympy import multiplicity as v
def a(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1) * (5**(v(5, n)+1)-1) // 8
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jul 08 2022

Crossrefs

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

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

Sequence in context: A279388 A292288 A347273 * A113957 A366148 A367171

Adjacent sequences: A355581 A355582 A355583 * A355585 A355586 A355587

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jul 08 2022

Status

approved