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A374163 a(1) = 1; for n>1 a(n) is the minimum value of k > 0 such that sigma^{k}(n)-1 is prime, if such a k exists; otherwise -1, where sigma^{k} is the k-th iteration of sigma=A000203.

%I A374163 #29 Jul 25 2024 14:48:21
%S A374163 1,1,1,2,1,1,1,2,2,1,1,4,1,1,1,2,1,4,1,1,1,5,1,1,2,1,2,3,1,1,1,2,1,1,
%T A374163 1,4,1,1,3,1,1,2,1,1,2,1,1,2,2,2,1,1,1,2,1,2,1,1,1,1,1,2,1,2,1,7,1,2,
%U A374163 2,7,1,3,1,1,2,1,2,1,1,2,8,2,1,1,1,1,2,1,1,1,3,1,1,7,2
%N A374163 a(1) = 1; for n>1 a(n) is the minimum value of k > 0 such that sigma^{k}(n)-1 is prime, if such a k exists; otherwise -1, where sigma^{k} is the k-th iteration of sigma=A000203.
%e A374163 For n=12, sigma^{4}(12)-1 = 360-1 = 359 is prime, and there is no positive k<4 such that sigma^{k}(12)-1 is prime, so a(12)=4.
%p A374163 sigma_iterate := proc (n, k)
%p A374163     local sigma_result, i:
%p A374163     sigma_result := n:
%p A374163     for i to k do
%p A374163         sigma_result := sigma(sigma_result)
%p A374163     end do:
%p A374163     return sigma_result
%p A374163 end proc:
%p A374163 find_min_k := proc (n)
%p A374163     local k, sigma_k_n, prime_candidate:
%p A374163     k := 0:
%p A374163     do
%p A374163         k := k+1:
%p A374163         sigma_k_n := sigma_iterate(n, k):
%p A374163         prime_candidate := sigma_k_n - 1:
%p A374163         if isprime(prime_candidate) then
%p A374163             return k
%p A374163         end if
%p A374163     end do
%p A374163 end proc:
%p A374163 map(find_min_k, [$ 2 .. 100]);
%t A374163 A374163[n_] := If[n==1, 1, Length[NestWhileList[DivisorSigma[1, #]&, n, !PrimeQ[# - 1]&, {2, 1}]] - 1]; Array[A374163, 100] (* _Paolo Xausa_, Jul 24 2024 *)
%o A374163 (PARI) a(n) = my(k=1, s=sigma(n)); while(!isprime(s-1), k++; s = sigma(s)); k; \\ _Michel Marcus_, Jun 29 2024
%Y A374163 Cf. A000203, A066421.
%K A374163 nonn
%O A374163 1,4
%A A374163 _Rafik Khalfi_, Jun 29 2024
%E A374163 Offset corrected by _N. J. A. Sloane_, Jul 25 2024