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.

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, 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, 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

Offset: 1

Author

Rafik Khalfi, Jun 29 2024

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A000203, A066421.

Programs

Maple

sigma_iterate := proc (n, k)
    local sigma_result, i:
    sigma_result := n:
    for i to k do
        sigma_result := sigma(sigma_result)
    end do:
    return sigma_result
end proc:
find_min_k := proc (n)
    local k, sigma_k_n, prime_candidate:
    k := 0:
    do
        k := k+1:
        sigma_k_n := sigma_iterate(n, k):
        prime_candidate := sigma_k_n - 1:
        if isprime(prime_candidate) then
            return k
        end if
    end do
end proc:
map(find_min_k, [$ 2 .. 100]);

Mathematica

A374163[n_] := If[n==1, 1, Length[NestWhileList[DivisorSigma[1, #]&, n, !PrimeQ[# - 1]&, {2, 1}]] - 1]; Array[A374163, 100] (* Paolo Xausa, Jul 24 2024 *)

PARI

a(n) = my(k=1, s=sigma(n)); while(!isprime(s-1), k++; s = sigma(s)); k; \\ Michel Marcus, Jun 29 2024

Extensions

Offset corrected by N. J. A. Sloane, Jul 25 2024