This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A083356 #26 Jul 11 2023 19:31:05
%S A083356 0,1,3,6,14,19,31,38,54,72,92,103,139,152,180,210,258,275,329,348,408,
%T A083356 450,494,517,613,663,715,769,853,882,1002,1033,1129,1195,1263,1333,
%U A083356 1513,1550,1626,1704,1864,1905,2073,2116,2248,2383,2475,2522,2762,2860
%N A083356 Total area of all incongruent integer-sided rectangles of area <= n.
%H A083356 Nick MacKinnon, <a href="https://www.jstor.org/stable/2695719">Problem 10883</a>, Amer. Math. Monthly, 108 (2001) 565; <a href="https://www.jstor.org/stable/3647894">solution</a> by John C. Cock, 110 (2003) 343-344.
%F A083356 a(n) = Sum_{k=1..n} k*ceiling(d(k)/2), where d(k)=A000005(k) is the number of divisors of k.
%F A083356 a(n) = Sum_{r=1..floor(sqrt(n))} r*(r+floor(n/r))*(floor(n/r)+1-r)/2.
%F A083356 a(n) = ( A143127(n) + A000330(floor(sqrt(n))) ) / 2. - _Max Alekseyev_, Jan 31 2012
%F A083356 a(n) ~ n^2 * log(n) / 4
%F A083356 G.f.: x*f'(x)/(1 - x), where f(x) = Sum_{k>=1} x^k^2/(1 - x^k). - _Ilya Gutkovskiy_, Apr 12 2017
%e A083356 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.
%t A083356 a[n_] := Sum[r(r+Floor[n/r])(Floor[n/r]+1-r), {r, 1, Floor[Sqrt[n]]}]/2
%o A083356 (Python)
%o A083356 from math import isqrt
%o A083356 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
%Y A083356 Cf. A083357, A143127
%Y A083356 Partial sums of A060872.
%K A083356 nonn,easy
%O A083356 0,3
%A A083356 _Dean Hickerson_, Apr 26 2003