A332533 - OEIS

%I #38 Jun 28 2024 05:05:15

%S 1,4,15,92,790,9384,137326,2397352,48428487,1111122360,28531183329,

%T 810554859732,25239592620853,854769763924104,31278135039463245,

%U 1229782938533709200,51702516368332126932,2314494592676172411516,109912203092257573556274,5518821052632117898282620

%N a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

%H Seiichi Manyama, <a href="/A332533/b332533.txt">Table of n, a(n) for n = 1..387</a>

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

%F a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.

%F a(n) ~ n^(n-1). - _Vaclav Kotesovec_, May 28 2021

%F a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - _Ridouane Oudra_, Mar 05 2023

%p seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # _Ridouane Oudra_, Mar 05 2023

%t Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]

%t Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]

%t Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]

%o (PARI) a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ _Michel Marcus_, Feb 16 2020

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ _Seiichi Manyama_, May 29 2021

%o (Magma)

%o A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;

%o [A332533(n): n in [1..40]]; // _G. C. Greubel_, Jun 27 2024

%o (SageMath)

%o def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))

%o [A332533(n) for n in range(1,41)] # _G. C. Greubel_, Jun 27 2024

%Y Cf. A024916, A268235, A308313, A308814, A319194, A320095.

%Y 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), A344816 (q=5), this sequence (q=n).

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Feb 16 2020