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

1,4

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).

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

Cf. A000203, A001065, A034090.

Sequence in context: A127739 A327142 A175394 * A257537 A219883 A219381

Adjacent sequences: A070315 A070316 A070317 * A070319 A070320 A070321

Keyword

easy,nonn,changed

Author

Benoit Cloitre, May 11 2002

Status

approved