A070318 a(n) = Max_{k=1..n} (sigma(k)-k) where sigma(k)-k is the sum of proper divisors of k.
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, 36, 36, 36, 42, 42, 42, 42, 42, 42, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Amiram Eldar, Plot of (1/n^2) * Sum_{i=1..n} a(i) for n = A034090(1..6524) (the positions of records; generated using the b-file at A034090).
- Amiram Eldar, Plot of (1/(n^2*log(log(n)))) * Sum_{i=1..n} a(i) for n = A034090(1..6524) (the positions of records; generated using the b-file at A034090).
Programs
-
Mathematica
FoldList[Max, Array[DivisorSigma[1, #] - # &, 100]] (* Amiram Eldar, Aug 04 2024 *)
-
PARI
lista(nmax) = {my(smax = -1); for(n = 1, nmax, smax = max(smax, sigma(n) - n); print1(smax, ", "));} \\ Amiram Eldar, Aug 04 2024
Formula
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]
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