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 A070318 #23 Jan 09 2025 02:49:27
%S A070318 0,1,1,3,3,6,6,7,7,8,8,16,16,16,16,16,16,21,21,22,22,22,22,36,36,36,
%T A070318 36,36,36,42,42,42,42,42,42,55,55,55,55,55,55,55,55,55,55,55,55,76,76,
%U A070318 76,76,76,76,76,76,76,76,76,76,108,108,108,108,108,108,108,108,108,108
%N A070318 a(n) = Max_{k=1..n} (sigma(k)-k) where sigma(k)-k is the sum of proper divisors of k.
%H A070318 Amiram Eldar, <a href="/A070318/b070318.txt">Table of n, a(n) for n = 1..10000</a>
%H A070318 Amiram Eldar, <a href="/A070318/a070318.jpg">Plot of (1/n^2) * Sum_{i=1..n} a(i) for n = A034090(1..6524)</a> (the positions of records; generated using the b-file at A034090).
%H A070318 Amiram Eldar, <a href="/A070318/a070318_1.jpg">Plot of (1/(n^2*log(log(n)))) * Sum_{i=1..n} a(i) for n = A034090(1..6524)</a> (the positions of records; generated using the b-file at A034090).
%F A070318 Limit_{n -> oo} (1/n^2) * Sum_{i=1..n} a(i) = C = 0.7... . [It seems that this limit in fact diverges to infinity; see the first plot in the links section. - _Amiram Eldar_, Aug 04 2024]
%F A070318 Conjecture: Limit_{n -> oo} (1/(n^2*log(log(n)))) * Sum_{i=1..n} a(i) = C = 0.7... . (see the second plot in the links section). - _Amiram Eldar_, Aug 04 2024
%t A070318 FoldList[Max, Array[DivisorSigma[1, #] - # &, 100]] (* _Amiram Eldar_, Aug 04 2024 *)
%o A070318 (PARI) lista(nmax) = {my(smax = -1); for(n = 1, nmax, smax = max(smax, sigma(n) - n); print1(smax, ", "));} \\ _Amiram Eldar_, Aug 04 2024
%Y A070318 Cf. A000203, A001065, A034090.
%K A070318 easy,nonn
%O A070318 1,4
%A A070318 _Benoit Cloitre_, May 11 2002