A057690 - OEIS

%I #32 Dec 12 2023 08:25:31

%S 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,

%T 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,

%U 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

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

%C See A057684 for definition.

%C 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

%H Michel Marcus, <a href="/A057690/b057690.txt">Table of n, a(n) for n = 2..10000</a>

%F a(n) = A023514(n)+1 if the cycle contains the number 1. - _Jon Maiga_, Jan 12 2021

%e 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.

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

%t 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 *)

%o (PARI) f(m, p) = {forprime(q=2, precprime(p-1), if (! (m % q), return (m/q));); m*p+1;}

%o 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

%o (Python)

%o from sympy import prime, primerange

%o def a(n):

%o     P = prime(n)

%o     x, plst, traj, seen = P, list(primerange(2, P)), [], set()

%o     while x not in seen:

%o         traj.append(x)

%o         seen.add(x)

%o         x = next((x//p for p in plst if x%p == 0), P*x+1)

%o     return len(traj) - traj.index(x)

%o print([a(n) for n in range(2, 107)]) # _Michael S. Branicky_, Dec 11 2023

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

%Y Cf. A023514.

%K nonn,nice,easy

%O 2,1

%A _N. J. A. Sloane_, Oct 20 2000

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

%E Corrected by _T. D. Noe_, Apr 02 2008