A344820 a(n) = Sum_{k=1..n} floor(n/k) * (-n)^(k-1).

1, 0, 9, -52, 520, -6636, 102984, -1864600, 38741463, -909081740, 23775986069, -685854111804, 21633935838489, -740800448012044, 27368368159530285, -1085102592823737200, 45957792326631241516, -2070863582899905915336, 98920982783031811482920, -4993219047619535240997780

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..387

Formula

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

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k/(1 + n*x^k).

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (-n)^(k-1) * x^k/(1 - x^k).

a(n) ~ -(-1)^n * n^(n-1). - Vaclav Kotesovec, Jun 05 2021

Mathematica

a[n_] := Sum[(-n)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 20] (* Amiram Eldar, May 29 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, n\k*(-n)^(k-1));
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-n)^(d-1)));
(Magma)
A344820:= func< n | (&+[Floor(n/k)*(-n)^(k-1): k in [1..n]]) >;
[A344820(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
(SageMath)
def A344820(n): return sum((n//k)*(-n)^(k-1) for k in range(1, n+1))
[A344820(n) for n in range(1, 41)] # G. C. Greubel, Jun 26 2024
(Python)
def A344820(n):
    c, j = 0, 1
    while j <= n:
        k = n//j
        m = n//k
        c += k*(n**(j-1) if j&1 else -n**(j-1))*(1+n**a if (a:=m-j+1)&1 else 1-n**a)
        j = m+1
    return c//(n+1) # Chai Wah Wu, May 11 2026

Crossrefs

Diagonal of A344824.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): this sequence (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), A344816 (q=5), A332533 (q=n).

Sequence in context: A278000 A159598 A279358 * A156544 A390152 A094793

Adjacent sequences: A344817 A344818 A344819 * A344821 A344822 A344823

Keyword

sign

Author

Seiichi Manyama, May 29 2021

Status

approved