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A057690
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Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.
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10
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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
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OFFSET
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2,1
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = A023514(n)+1 if the cycle contains the number 1. - Jon Maiga, Jan 12 2021
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057691.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000
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STATUS
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approved
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