cp's OEIS Frontend

A355888 a(n) = Sum_{k=1..n} k! * floor(n/k).

%I A355888 #18 Aug 11 2025 06:32:58
%S A355888 1,4,11,38,159,888,5929,46276,409163,4038086,43954887,522957240,
%T A355888 6749978041,93928274284,1401602642411,22324392570758,378011820666759,
%U A355888 6780385526758368,128425485935590369,2561327494115859316,53652269665825304363,1177652997443472901166
%N A355888 a(n) = Sum_{k=1..n} k! * floor(n/k).
%F A355888 a(n) = Sum_{k=1..n} Sum_{d|k} d!.
%F A355888 G.f.: (1/(1-x)) * Sum_{k>0} k! * x^k/(1 - x^k).
%F A355888 a(n) ~ n!. - _Vaclav Kotesovec_, Aug 11 2025
%t A355888 Table[Sum[k!*Floor[n/k], {k,1,n}], {n,1,25}] (* _Vaclav Kotesovec_, Aug 11 2025 *)
%o A355888 (PARI) a(n) = sum(k=1, n, n\k*k!);
%o A355888 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, d!));
%o A355888 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*x^k/(1-x^k))/(1-x))
%o A355888 (Python)
%o A355888 from math import factorial
%o A355888 def A355888(n): return factorial(n)+n+sum(factorial(k)*(n//k) for k in range(2,n)) if n>1 else 1 # _Chai Wah Wu_, Jul 21 2022
%Y A355888 Partial sums of A062363.
%Y A355888 Cf. A006218, A024916.
%K A355888 nonn
%O A355888 1,2
%A A355888 _Seiichi Manyama_, Jul 20 2022