A380609 - OEIS

%I #23 Feb 16 2025 19:36:35

%S 2,17,31,37,67,71,73,97,103,137,149,157,181,199,211,227,241,269,283,

%T 313,337,367,379,409,487,541,563,577,587,607,617,643,661,769,787,857,

%U 877,907,929,937,977,997,1039,1093,1151,1187,1237,1453,1543,1567,1579,1621

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

%C 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.

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

%e 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).

%p gpf:= n -> max(numtheory:-factorset(n)):

%p filter:= proc(n) local S,t,x;

%p   t:= gpf(2*n+1);

%p   if t = n then return false fi;

%p   S:= {n,t};

%p   x:= t;

%p   do

%p     x:= gpf(2*x+1);

%p     if member(x,S) then return (x = t) fi;

%p     S:= S union {x};

%p   od;

%p end proc:

%p select(filter, [seq(ithprime(i),i=1..1000)]); # _Robert Israel_, Feb 03 2025

%Y Cf. A006530, A076565, A287865.

%K nonn

%O 1,1

%A _Johannes M.V.A. Koelman_, Jan 28 2025