A340967 a(n) is the number of iterations of the map x -> n mod sopfr(x) starting with n to reach 0 or 1, where sopfr = A001414.

0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 1, 1, 2, 1, 2, 1, 4, 1, 3, 2, 4, 1, 3, 1, 1, 1, 2, 3, 3, 3, 2, 1, 4, 4, 3, 1, 3, 1, 4, 1, 3, 1, 2, 2, 2, 4, 1, 1, 5, 3, 2, 4, 4, 1, 1, 1, 4, 4, 2, 4, 2, 1, 4, 2, 1, 1, 1, 1, 3, 3, 3, 3, 3, 1, 2, 3, 3, 1, 1, 3, 4, 5, 2, 1, 3, 3, 3, 3, 5, 4, 2, 1, 2, 2

Offset: 1

Author

J. M. Bergot and Robert Israel, Jan 31 2021

Keywords

• nonn

Comments

If n is prime, or n is in A164643, then a(n) = 1.

Examples

			a(12) = 3 because 12 mod (2+2+3) = 5, 12 mod 5 = 2 and 12 mod 2 = 0 (3 iterations).
a(54) = 5 because 54 mod (2+3+3+3) = 10, 54 mod (2+5) = 6, 54 mod 5 = 4, 54 mod (2+2) = 2, and 54 mod 2 = 0 (5 iterations).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A164643, A340969.

Programs

Maple

sopfr:= proc(n) local t;
  add(t[1]*t[2], t = ifactors(n)[2])
end proc:
f:= proc(n) local x,k;
  x:= n;
  for k from 1 do x:= n mod sopfr(x); if x <= 1 then return k fi od;
end proc:
f(1):= 0:
map(f, [$1..200]);

Python

from sympy import factorint
def A340967(n):
    c, x = 0, n
    while x > 1:
        c += 1
        x = n % sum(p*e for p, e in factorint(x).items())
    return c # Chai Wah Wu, Feb 01 2021