A344815 a(n) = Sum_{k=1..n} floor(n/k) * 4^(k-1).

1, 6, 23, 92, 349, 1394, 5491, 21944, 87497, 349902, 1398479, 5593892, 22371109, 89484074, 357919803, 1431678080, 5726645377, 22906581142, 91626057879, 366504227292, 1466015859181, 5864063418866, 23456249463283, 93824997852744, 375299974563657, 1501199898183502

Offset

1,2

Comments

Partial sums of A339684.

Links

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

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} 4^(d-1).

G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 4*x^k).

G.f.: (1/(1 - x)) * Sum_{k>=1} 4^(k-1) * x^k/(1 - x^k).

a(n) ~ 4^n / 3. - Vaclav Kotesovec, Jun 05 2021

a(n) = (1/3) * Sum_{k=1..n} (4^floor(n/k) - 1). - Ridouane Oudra, Feb 16 2023

Maple

seq(add(4^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Feb 16 2023

Mathematica

a[n_] := Sum[4^(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*4^(k-1));
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, 4^(d-1)));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-4*x^k))/(1-x))
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 4^(k-1)*x^k/(1-x^k))/(1-x))
(Magma)
A344815:= func< n | (&+[Floor(n/j)*4^(j-1): j in [1..n]]) >;
[A344815(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024
(SageMath)
def A344815(n): return sum((n//j)*4^(j-1) for j in range(1, n+1))
[A344815(n) for n in range(1, 41)] # G. C. Greubel, Jun 27 2024

Crossrefs

Column k=4 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), this sequence (q=4), A344816 (q=5), A332533 (q=n).

Sequence in context: A013261 A013265 A228703 * A220372 A321715 A038383

Adjacent sequences: A344812 A344813 A344814 * A344816 A344817 A344818

Keyword

nonn

Author

Seiichi Manyama, May 29 2021

Status

approved