A344816 a(n) = Sum_{k=1..n} floor(n/k) * 5^(k-1).

1, 7, 33, 164, 790, 3946, 19572, 97828, 488479, 2442235, 12207861, 61039267, 305179893, 1525898649, 7629414925, 38147071306, 190734961932, 953674808838, 4768372074464, 23841860356470, 119209292012746, 596046459981502, 2980232250997128, 14901161254984784

Offset

1,2

Comments

Partial sums of A339685.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} 5^(d-1).

G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 5*x^k).

G.f.: (1/(1 - x)) * Sum_{k>=1} 5^(k-1) * x^k/(1 - x^k).

a(n) ~ 5^n / 4. - Vaclav Kotesovec, Jun 05 2021

a(n) = (1/4) * Sum_{k=1..n} (5^floor(n/k) - 1). - Ridouane Oudra, Mar 05 2023

Maple

seq(add(5^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023

Mathematica

a[n_] := Sum[5^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, n\k*5^(k-1));
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, 5^(d-1)));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-5*x^k))/(1-x))
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 5^(k-1)*x^k/(1-x^k))/(1-x))
(Magma)
A344816:= func< n | (&+[Floor(n/k)*5^(k-1): k in [1..n]]) >;
[A344816(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
(SageMath)
def A344816(n): return sum((n//k)*5^(k-1) for k in range(1, n+1))
[A344816(n) for n in range(1, 41)] # G. C. Greubel, Jun 26 2024

Crossrefs

Column k=5 of A344821.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), this sequence (q=5), A332533 (q=n).

Sequence in context: A304278 A155603 A282991 * A295270 A275860 A054256

Adjacent sequences: A344813 A344814 A344815 * A344817 A344818 A344819

Keyword

nonn

Author

Seiichi Manyama, May 29 2021

Status

approved