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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 29 2021
STATUS
approved