A057689 Maximal term in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if no such term exists.

16, 66, 50, 672, 20372, 494, 36918, 1404, 12210, 4248, 5070, 1682, 1850, 2210, 35882, 102720, 94484303672, 30084, 178992, 5330, 246560, 6890, 294253314, 8416400, 515202, 134004, 2810784, 2810883506682183650, 377198408, 320168, 3742442930913284, 257150

Offset

2,1

Comments

The 'Px+1' map is: x -> (x/p if p = least prime factor(x) < P, else P*x + 1).

Links

Michael S. Branicky, Table of n, a(n) for n = 2..10001 (terms 2..1000 from T. D. Noe)

Example

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.

Mathematica

Px1[p_, n_]:=Catch[For[i=1, i<PrimePi[p], i++, If[Divisible[n, Prime[i]], Throw[n/Prime[i]]]]; p*n+1];
With[{nmax=50}, Table[Max[NestWhileList[Px1[Prime[n], #]&, Prime[n], UnsameQ, All]], {n, 2, nmax}]] (* Paolo Xausa, Dec 11 2023 *)

Prog

(Python)
from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, seen = P, list(primerange(2, P)), set()
    while x > 1 and x not in seen:
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return max(seen)
print([a(n) for n in range(2, 32)]) # Michael S. Branicky, Dec 11 2023
(PARI) apply( {A057689(n)=my(m=oo, p=n=prime(n), M=n); until(n==m, if(n<m, m=n, n>M, M=n); forprime(q=2, p-1, if(n%q==0, n\=q; next(2))); n=n*p+1); M}, [2..33]) \\ M. F. Hasler, Feb 13 2026

Crossrefs

Cf. A057690, A057691 (cycle and transition length for Px+1).

Cf. A033478, A057688, A057684, A057685, A057686, A057687 (trajectory of P under Px+1 for P = 3, 5, 13, 19, 23, 29).

Cf. A057614, A057522, A057446, A057534, A057216 (other trajectories under 11x+1, 13x+1 and 17x+1).

Cf. A020639 (least prime factor).

Sequence in context: A119285 A253673 A041494 * A045024 A043153 A039330

Adjacent sequences: A057686 A057687 A057688 * A057690 A057691 A057692

Keyword

nonn,nice

Author

N. J. A. Sloane, Oct 20 2000

Extensions

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

Status

approved