A057690 Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

3, 3, 4, 4, 3, 4, 4, 5, 4, 6, 3, 4, 4, 6, 5, 5, 3, 4, 6, 3, 6, 5, 5, 4, 4, 5, 6, 4, 4, 8, 5, 4, 5, 5, 5, 3, 4, 6, 4, 6, 4, 8, 3, 5, 6, 4, 7, 5, 4, 5, 7, 4, 6, 4, 6, 6, 6, 3, 12, 4, 5, 5, 6, 3, 4, 4, 4, 5, 5, 4, 7, 6, 4, 5, 9, 5, 3, 4, 4, 6, 3, 8, 4, 6, 5, 6, 3, 5, 6, 6, 8, 5, 5, 6, 7, 5, 5, 4, 3, 4, 5, 5, 5, 5, 4

Offset: 2

Author

N. J. A. Sloane, Oct 20 2000

Keywords

• nonn
• nice
• easy

Comments

See A057684 for definition.

Note that not all cycles for the iteration starting with p contain the number 1; a(60), for the prime 281, is the first example of this. Its iterates are: 281, 78962, 39481, 3037, 853398, 426699, 142233, 47411, 6773, 521, 146402, 73201, 1031, 289712, 144856, 72428, 36214, 18107, 953, 267794, 133897, with the last 12 terms cycling. Another example is provided by 2543, the 372nd prime. - T. D. Noe, Apr 02 2008

Examples

			For n=4, 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

• Michel Marcus, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057691.

Cf. A023514.

Programs

Mathematica

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

PARI

f(m, p) = {forprime(q=2, precprime(p-1), if (! (m % q), return (m/q));); m*p+1;}
a(n) = {my(p=prime(n), x=p, list = List()); listput(list, x); while (1, x = f(x, p); for (i=1, #list, if (x == list[i], return (#list - i + 1));); listput(list, x););} \\ Michel Marcus, Jan 12 2021

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 len(traj) - traj.index(x)
print([a(n) for n in range(2, 107)]) # Michael S. Branicky, Dec 11 2023

Formula

a(n) = A023514(n)+1 if the cycle contains the number 1. - Jon Maiga, Jan 12 2021

Extensions

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

Corrected by T. D. Noe, Apr 02 2008