A361685 Number of iterations of sopf until reaching a prime.

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 3, 1, 2, 0, 2, 0, 1, 3, 1, 2, 1, 0, 3, 2, 1, 0, 2, 0, 1, 2, 2, 0, 1, 1, 1, 2, 3, 0, 1, 2, 2, 2, 1, 0, 2, 0, 4, 2, 1, 2, 2, 0, 1, 4, 3, 0, 1, 0, 3, 2, 3, 2, 2, 0, 1, 1, 1, 0, 2, 2, 3, 2, 1, 0, 2, 2, 2, 2, 2, 2, 1, 0, 2, 3, 1, 0

Offset: 2

Author

J. W. Montgomery, Mar 29 2023

Keywords

• nonn

Examples

			a(15) = 2 because 15 is not prime, sopf(15) = 8 is not prime, and sopf^2(15) = sopf(8) = 2 is prime.
a(16) = 1 because 16 is not prime and sopf(16) = 2 is prime.
a(17) = 0 because 17 is prime.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..65537

Crossrefs

Cf. A008472 (sopf), A321944.

Programs

MATLAB

for n=2:101
    s = n;
    c = 0;
    while ~isprime(s)
        s = sum(unique(factor(s)));
        c = c + 1;
    end
    a(n) = c;
end

PARI

A008472(n) = vecsum(factor(n)[, 1]);
A361685(n) = for(k=0,oo,if(isprime(n),return(k)); n = A008472(n)); \\ Antti Karttunen, Jan 28 2025

Formula

For n >= 2, a(n) = min{m : sopf^m(n) is prime} where sopf^m indicates m iterations of sopf, the sum of the prime factors function.

a(n) = A321944(n) - 1. - Rémy Sigrist, Mar 29 2023