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

Author

Joseph L. Pe, Dec 26 2001

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A099433, A099434.

Programs

Mathematica

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

PARI

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

Extensions

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

Description clarified by Antti Karttunen, Nov 07 2017