A339370 - OEIS

%I #20 Oct 27 2023 20:38:10

%S 0,1,4,11,19,36,50,76,102,138,165,227,262,318,381,460,510,614,672,791,

%T 889,990,1064,1249,1353,1477,1610,1790,1891,2133,2244,2455,2626,2798,

%U 2983,3312,3452,3649,3857,4198,4356,4715,4883,5190,5514,5763,5949,6446,6686,7045

%N a(n) = Sum_{k=1..floor(n/2)} (n-k) * floor((n-k)/k).

%C Total area of all y X z rectangles such that x + y = n, with x and y integers, 0 < x <= y and z = floor(y/x). - _Wesley Ivan Hurt_, Dec 21 2020

%F a(n) ~ n^2*(log(n) + 2*EulerGamma - Pi^2/12 - 3/2). - _Rok Cestnik_, Dec 20 2020

%F a(n) = n*A002541(n) - A153485(n). - _Vaclav Kotesovec_, Dec 21 2020

%t Table[Sum[(n - k)*Floor[(n - k)/k], {k, Floor[n/2]}], {n, 60}]

%o (PARI) a(n) = sum(k=1, n\2, (n-k) * ((n-k)\k)); \\ _Michel Marcus_, Dec 02 2020

%o (Python)

%o from math import isqrt

%o def A339370(n): return n*(1-n)+(s:=isqrt(n))**2*(s+1-(n<<1))-sum((q:=n//k)*((k-(n<<1)<<1)+q+1) for k in range(1,s+1))>>1 # _Chai Wah Wu_, Oct 27 2023

%Y Cf. A002541, A153485.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Dec 01 2020