A066421 a(n) = least k > 0 such that sigma^(k)(n) + 1 is prime, if such k exists; otherwise 0, where sigma^(k) denotes application of sigma k times.

1, 2, 1, 5, 1, 1, 4, 3, 4, 1, 1, 1, 3, 2, 2, 5, 1, 5, 2, 1, 4, 1, 2, 1, 5, 1, 1, 4, 1, 1, 4, 3, 9, 4, 9, 2, 2, 1, 4, 3, 1, 1, 9, 8, 1, 1, 9, 8, 5, 4, 1, 5, 4, 3, 1, 3, 4, 3, 1, 4, 2, 1, 2, 4, 8, 3, 2, 1, 1, 3, 1, 2, 3, 2, 8, 2, 1, 4, 4, 3, 4, 1, 8, 7, 1, 2, 3, 1, 3, 2, 1, 4, 3, 3, 3, 4, 5, 4, 1, 2

Offset

1,2

Comments

Does the orbit of the arithmetical dynamical system f(n) = sigma(n) contain one less than a prime, for every initial point n? That is to say, is a(n) nonzero for every n?

a(n) > 0 for all n < 36090. If a(36090) > 0, it is > 159. - Gabriel Cunningham (gcasey(AT)mit.edu), Oct 15 2004

a(n) > 0 for all n <= 675000 and a(36090)=291. - Sean A. Irvine, Oct 15 2023

Links

Antti Karttunen, Table of n, a(n) for n = 1..36089

Example

sigma(sigma(sigma(8))) + 1 = sigma(sigma(15)) + 1 = sigma(24) + 1 = 60 + 1 = 61, a prime; hence a(8) = 3.

Mathematica

A066421[n_]:=Length[NestWhileList[DivisorSigma[1, #]&, DivisorSigma[1, n], !PrimeQ[#+1]&]]; Array[A066421, 100] (* Paolo Xausa, Oct 16 2023 *)

Prog

(PARI) A066421(n) = { my(k=1, s=sigma(n)); while(!isprime(1+s), k++; s = sigma(s)); k; }; \\ Antti Karttunen, Nov 07 2017

Crossrefs

Cf. A099433, A099434.

Sequence in context: A348494 A256541 A342919 * A369526 A206563 A299779

Adjacent sequences: A066418 A066419 A066420 * A066422 A066423 A066424

Keyword

nonn

Author

Joseph L. Pe, Dec 26 2001

Extensions

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 15 2004

Description clarified by Antti Karttunen, Nov 07 2017

Status

approved