cp's OEIS Frontend

A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

%I A309125 #16 Mar 27 2022 03:09:09
%S A309125 1,2,3,8,9,10,11,16,26,27,28,33,34,35,36,57,58,68,69,74,75,76,77,82,
%T A309125 108,109,119,124,125,126,127,148,149,150,151,201,202,203,204,209,210,
%U A309125 211,212,217,227,228,229,250,300,326,327,332,333,343,344,349,350,351,352,357,358,359,369,454,455,456
%N A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...
%C A309125 Partial sums of A035316.
%H A309125 Seiichi Manyama, <a href="/A309125/b309125.txt">Table of n, a(n) for n = 1..10000</a>
%H A309125 Vaclav Kotesovec, <a href="/A309125/a309125.jpg">Plot of a(n)/n^(3/2) for n = 1..10000</a>
%F A309125 G.f.: (1/(1 - x)) * Sum_{k>=1} k^2 * x^(k^2)/(1 - x^(k^2)).
%F A309125 a(n) ~ zeta(3/2)*n^(3/2)/3 - n/2. - _Vaclav Kotesovec_, Aug 30 2021
%t A309125 Table[Sum[k^2 Floor[n/k^2], {k, 1, n}], {n, 1, 66}]
%t A309125 nmax = 66; CoefficientList[Series[1/(1 - x) Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}], {x, 0, nmax}], x] // Rest
%o A309125 (PARI) a(n) = sum(k=1, n, k^2*(n\k^2)); \\ _Seiichi Manyama_, Aug 30 2021
%Y A309125 Cf. A013936, A024916, A035316, A309126, A309127.
%K A309125 nonn
%O A309125 1,2
%A A309125 _Ilya Gutkovskiy_, Jul 13 2019