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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.

%I A066421 #16 Oct 16 2023 04:32:38
%S A066421 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,
%T A066421 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,
%U A066421 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
%N 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.
%C A066421 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?
%C A066421 a(n) > 0 for all n < 36090. If a(36090) > 0, it is > 159. - Gabriel Cunningham (gcasey(AT)mit.edu), Oct 15 2004
%C A066421 a(n) > 0 for all n <= 675000 and a(36090)=291. - _Sean A. Irvine_, Oct 15 2023
%H A066421 Antti Karttunen, <a href="/A066421/b066421.txt">Table of n, a(n) for n = 1..36089</a>
%e A066421 sigma(sigma(sigma(8))) + 1 = sigma(sigma(15)) + 1 = sigma(24) + 1 = 60 + 1 = 61, a prime; hence a(8) = 3.
%t A066421 A066421[n_]:=Length[NestWhileList[DivisorSigma[1,#]&,DivisorSigma[1,n],!PrimeQ[#+1]&]];Array[A066421,100] (* _Paolo Xausa_, Oct 16 2023 *)
%o A066421 (PARI) A066421(n) = { my(k=1,s=sigma(n)); while(!isprime(1+s),k++;s = sigma(s)); k; }; \\ _Antti Karttunen_, Nov 07 2017
%Y A066421 Cf. A099433, A099434.
%K A066421 nonn
%O A066421 1,2
%A A066421 _Joseph L. Pe_, Dec 26 2001
%E A066421 More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 15 2004
%E A066421 Description clarified by _Antti Karttunen_, Nov 07 2017