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A070318
a(n) = Max_{k=1..n} (sigma(k)-k) where sigma(k)-k is the sum of proper divisors of k.
4
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
1,4
LINKS
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).
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
MATHEMATICA
FoldList[Max, Array[DivisorSigma[1, #] - # &, 100]] (* Amiram Eldar, Aug 04 2024 *)
PROG
(PARI) lista(nmax) = {my(smax = -1); for(n = 1, nmax, smax = max(smax, sigma(n) - n); print1(smax, ", ")); } \\ Amiram Eldar, Aug 04 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, May 11 2002
STATUS
approved