A057691 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

See A057684 for definition.

LINKS

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

EXAMPLE

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.

MATHEMATICA

Px1[p_, n_]:=Catch[For[i=1, i<PrimePi[p], i++, If[Divisible[n, Prime[i]], Throw[n/Prime[i]]]]; p*n+1];

Module[{nmax=100, m}, Table[FirstPosition[m=NestWhileList[Px1[Prime[n], #]&, Prime[n], UnsameQ, All], Last[m]][[1]]-1, {n, 2, nmax}]] (* Paolo Xausa, Dec 11 2023 *)

PROG

(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