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
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
KEYWORD
nonn,nice,easy
AUTHOR
N. J. A. Sloane, Oct 20 2000
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000
Corrected by T. D. Noe, Apr 02 2008
STATUS
approved