A364223 Expansion of Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k))^2.

1, 2, 3, 4, 10, 6, 7, 8, 9, 20, 11, 12, 13, 14, 30, 16, 17, 18, 19, 40, 21, 22, 23, 24, 75, 26, 27, 28, 29, 60, 31, 32, 33, 34, 70, 36, 37, 38, 39, 80, 41, 42, 43, 44, 90, 46, 47, 48, 49, 150, 51, 52, 53, 54, 110, 56, 57, 58, 59, 120, 61, 62, 63, 64, 130, 66, 67, 68, 69, 140, 71, 72, 73, 74, 225

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

a(n) = n * A055457(n).

If n == 0 (mod 5), a(n) = n + 5 * a(n/5) otherwise a(n) = n.

From Amiram Eldar, Jul 14 2023: (Start)

Multiplicative with a(5^e) = (e+1)*5^e and a(p^e) = p*e if p != 5.

Dirichlet g.f.: (5^s/(5^s-5)) * zeta(s-1).

Sum_{k=1..n} a(k) ~ (5/8)*n^2. (End)

Maple

a:= n-> (1+padic[ordp](n, 5))*n:
seq(a(n), n=1..75);  # Alois P. Heinz, Nov 25 2025

Mathematica

a[n_] := n * (IntegerExponent[n, 5] + 1); Array[a, 100] (* Amiram Eldar, Jul 14 2023 *)

Prog

(PARI) a(n) = n*(valuation(n, 5)+1);

Crossrefs

Cf. A055457, A112765, A359100.

Cf. A091512, A364222, A364224.

Sequence in context: A129920 A356332 A065634 * A394533 A087548 A353959

Adjacent sequences: A364220 A364221 A364222 * A364224 A364225 A364226

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Jul 14 2023

Status

approved