A057690 - OEIS

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

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

OFFSET

2,1

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

LINKS

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

FORMULA

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

EXAMPLE

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.

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

PROG

(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

CROSSREFS