A083356 Total area of all incongruent integer-sided rectangles of area <= n.

0, 1, 3, 6, 14, 19, 31, 38, 54, 72, 92, 103, 139, 152, 180, 210, 258, 275, 329, 348, 408, 450, 494, 517, 613, 663, 715, 769, 853, 882, 1002, 1033, 1129, 1195, 1263, 1333, 1513, 1550, 1626, 1704, 1864, 1905, 2073, 2116, 2248, 2383, 2475, 2522, 2762, 2860

Offset: 0

Author

Dean Hickerson, Apr 26 2003

Keywords

• nonn
• easy

Examples

			a(5)=19, the rectangles being 1 X 1, 1 X 2, 1 X 3, 1 X 4, 1 X 5 and 2 X 2.
		

Links

• Nick MacKinnon, Problem 10883, Amer. Math. Monthly, 108 (2001) 565; solution by John C. Cock, 110 (2003) 343-344.

Crossrefs

Cf. A083357, A143127

Partial sums of A060872.

Programs

Mathematica

a[n_] := Sum[r(r+Floor[n/r])(Floor[n/r]+1-r), {r, 1, Floor[Sqrt[n]]}]/2

Python

from math import isqrt
def A083356(n): return (k:=isqrt(n))*(k+1)*(2+4*k-3*k*(k+1))//24+sum(i*(m:=n//i)*(1+m)>>1 for i in range(1,k+1)) # Chai Wah Wu, Jul 11 2023

Formula

a(n) = Sum_{k=1..n} k*ceiling(d(k)/2), where d(k)=A000005(k) is the number of divisors of k.

a(n) = Sum_{r=1..floor(sqrt(n))} r*(r+floor(n/r))*(floor(n/r)+1-r)/2.

a(n) = ( A143127(n) + A000330(floor(sqrt(n))) ) / 2. - Max Alekseyev, Jan 31 2012

a(n) ~ n^2 * log(n) / 4

G.f.: x*f'(x)/(1 - x), where f(x) = Sum_{k>=1} x^k^2/(1 - x^k). - Ilya Gutkovskiy, Apr 12 2017