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%I #20 Dec 12 2023 08:25:37
%S 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,
%T 18,33,288,47,29,317,14,12,61,60,6,16,10,5,14,46,5,6,15,105,4,11,48,
%U 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
%N 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.
%C See A057684 for definition.
%H Michael S. Branicky, <a href="/A057691/b057691.txt">Table of n, a(n) for n = 2..10001</a> (terms 2..1000 from T. D. Noe)
%e 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.
%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[FirstPosition[m=NestWhileList[Px1[Prime[n],#]&,Prime[n],UnsameQ,All],Last[m]][[1]]-1,{n,2,nmax}]] (* _Paolo Xausa_, Dec 11 2023 *)
%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 traj.index(x)
%o print([a(n) for n in range(2, 82)]) # _Michael S. Branicky_, Dec 11 2023
%Y Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057690.
%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
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