A057691 Number of terms before entering cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

5, 13, 4, 10, 25, 11, 68, 14, 39, 34, 9, 4, 5, 5, 16, 16, 234, 23, 16, 5, 11, 5, 63, 116, 18, 18, 33, 288, 47, 29, 317, 14, 12, 61, 60, 6, 16, 10, 5, 14, 46, 5, 6, 15, 105, 4, 11, 48, 44, 5, 6, 10, 5, 55, 15, 14, 25, 17, 9, 16, 6, 7, 26, 5, 33, 46, 5, 16, 23, 13, 15, 11, 16, 14, 11

Offset: 2

Author

N. J. A. Sloane, Oct 20 2000

Keywords

• nonn
• nice
• easy

Comments

See A057684 for definition.

Examples

			For n=3, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 2..10001 (terms 2..1000 from T. D. Noe)

Crossrefs

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057690.

Programs

Mathematica

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

Python

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, traj, seen = P, list(primerange(2, P)), [], set()
    while x not in seen:
        traj.append(x)
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return traj.index(x)
print([a(n) for n in range(2, 82)]) # Michael S. Branicky, Dec 11 2023

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000