A338991 a(n) = Sum_{k=1..floor(n/2)} (n-2*k) * floor((n-k)/k).

0, 0, 2, 6, 13, 24, 37, 56, 78, 106, 132, 178, 212, 258, 312, 376, 425, 508, 565, 662, 749, 836, 909, 1058, 1156, 1264, 1384, 1536, 1636, 1836, 1946, 2126, 2282, 2434, 2606, 2880, 3019, 3194, 3385, 3676, 3833, 4138, 4305, 4572, 4863, 5086, 5271, 5692, 5924, 6240

Offset

1,3

Comments

Total area of all rectangles with dimensions (y-x) X (z) where x and y are integers such that x + y = n, 0 < x <= y, and z = floor(y/x).

Links

Table of n, a(n) for n=1..50.

Formula

From Vaclav Kotesovec, Jun 24 2021: (Start)

a(n) = n + n*A006218(n) - 2*A024916(n).

a(n) ~ (log(n) + 2*gamma - Pi^2/6 - 1)*n^2, where gamma is the Euler-Mascheroni constant A001620. (End)

Mathematica

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

Prog

(Python)
from math import isqrt
def A338991(n): return ((s:=isqrt(n))+1)*(n*(1-s)+s**2)-sum((q:=n//k)*((k-n<<1)+q+1) for k in range(1, s+1)) # Chai Wah Wu, Oct 23 2023

Crossrefs

Cf. A002541, A153485, A279847, A339370.

Sequence in context: A011891 A370880 A184533 * A178532 A003600 A283551

Adjacent sequences: A338988 A338989 A338990 * A338992 A338993 A338994

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 21 2020

Status

approved