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a(n) = Max_{k=1..n} (sigma(k)-k) where sigma(k)-k is the sum of proper divisors of k.
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%I #19 Aug 04 2024 03:02:36

%S 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 36,36,36,42,42,42,42,42,42,55,55,55,55,55,55,55,55,55,55,55,55,76,76,

%U 76,76,76,76,76,76,76,76,76,76,108,108,108,108,108,108,108,108,108,108

%N a(n) = Max_{k=1..n} (sigma(k)-k) where sigma(k)-k is the sum of proper divisors of k.

%H Amiram Eldar, <a href="/A070318/b070318.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 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 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 FoldList[Max, Array[DivisorSigma[1, #] - # &, 100]] (* _Amiram Eldar_, Aug 04 2024 *)

%o (PARI) lista(nmax) = {my(smax = -1); for(n = 1, nmax, smax = max(smax, sigma(n) - n); print1(smax, ", "));} \\ _Amiram Eldar_, Aug 04 2024

%Y Cf. A000203, A001065, A034090.

%K easy,nonn

%O 1,4

%A _Benoit Cloitre_, May 11 2002