A380609 Primes a single step away from a cycle under the mapping p-> gpf(2*p+1).

2, 17, 31, 37, 67, 71, 73, 97, 103, 137, 149, 157, 181, 199, 211, 227, 241, 269, 283, 313, 337, 367, 379, 409, 487, 541, 563, 577, 587, 607, 617, 643, 661, 769, 787, 857, 877, 907, 929, 937, 977, 997, 1039, 1093, 1151, 1187, 1237, 1453, 1543, 1567, 1579, 1621

Offset: 1

Author

Johannes M.V.A. Koelman, Jan 28 2025

Keywords

• nonn

Comments

The cycle that gets entered consists of the primes in A287865. It appears that the mapping p -> gpf(2*p+1) produces no other cycles.

Conjecture: under repeated mapping all primes ultimately enter the same cycle.

Examples

			Prime 2 is in the sequence as it maps to 5. And so is 17 as it maps to 7.  The primes 3, 5, 7, 11, 13, 19, 23 and 47 are not included, as they are part of the cycle itself (and hence considered zero iterations away from the cycle).
		

Crossrefs

Cf. A006530, A076565, A287865.

Programs

Maple

gpf:= n -> max(numtheory:-factorset(n)):
filter:= proc(n) local S,t,x;
  t:= gpf(2*n+1);
  if t = n then return false fi;
  S:= {n,t};
  x:= t;
  do
    x:= gpf(2*x+1);
    if member(x,S) then return (x = t) fi;
    S:= S union {x};
  od;
end proc:
select(filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 03 2025