cp's OEIS Frontend

A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

%I A333194 #20 May 26 2020 06:27:42
%S A333194 1,2,4,4,8,8,11,11,19,16,21,21,30,30,37,29,45,45,51,51,66,56,67,67,88,
%T A333194 83,96,84,105,105,112,112,144,130,147,135,159,159,178,162,197,197,208,
%U A333194 208,241,209,232,232,277,270,290,270,309,309,324,308,357,335,364,364
%N A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.
%F A333194 G.f.: (x/(1 - x)) * (1/(1 - x)^2 - Sum_{k>=1} k * x^k / (1 + x^k)).
%F A333194 a(n) = n*(n + 1)/2 - Sum_{k=1..n-1} A000593(k).
%F A333194 a(n) = A000217(n) - A078471(n-1).
%p A333194 b:= n-> add(d, d=select(x-> x::odd, numtheory[divisors](n))):
%p A333194 a:= proc(n) option remember; n+`if`(n<2, 0, a(n-1))-b(n-1) end:
%p A333194 seq(a(n), n=1..60);  # _Alois P. Heinz_, May 25 2020
%t A333194 Table[Sum[Mod[Ceiling[n/k], 2] k, {k, 1, n}], {n, 1, 60}]
%t A333194 Table[n (n + 1)/2 - Sum[DivisorSum[k, (-1)^(k/# + 1) # &], {k, 1, n - 1}], {n, 1, 60}]
%t A333194 nmax = 60; CoefficientList[Series[x/(1 - x) (1/(1 - x)^2 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest
%o A333194 (PARI) a(n) = sum(k=1, n, (ceil(n/k) % 2)*k); \\ _Michel Marcus_, May 26 2020
%Y A333194 Cf. A000217, A000593, A078471, A120885, A330926, A332490.
%K A333194 nonn
%O A333194 1,2
%A A333194 _Ilya Gutkovskiy_, May 25 2020