%I #26 Jul 30 2023 09:22:55
%S 1,4,5,3,3,5,3,8,6,4,21,17,7,7,5,5,22,24,20,18,18,16,8,6,8,6,29,23,27,
%T 23,23,21,19,19,17,21,17,15,7,7,9,60,9,9,7,30,28,26,24,26,24,24,28,24,
%U 22,20,20,22,20,18,20,18,20,18,18,16,14,12,10,12,10,61,59,55,12,10,8,31
%N Number of iterations of A138754 before reaching a number for the second time, when starting with n.
%C This is a variation of A138752, giving the number of iterations of A138754 needed to get any number for the second time, while A138752 stops counting somehow arbitrarily at 1=primepi(2) or 4=primepi(7).
%C The map A138754 is a variation of the Collatz map where parity of the integers is replaced by p mod 3 for the primes.
%C For the Collatz map, we have the only fixed point 0=f(0) and all other numbers seem to end up in the cycle 1->4->2->1.
%C Here the only fixed point is 1=A138754(1) and all other numbers seem to end up in the cycle 4 -> 7 -> 5 -> 4 (corresponding to primes 7 -> 17 -> 11 -> 7).
%C Depending on which number among primepi({2,7,11,17}) is reached first, A138753(n) = A138752(n)+1,+3,+2 resp. +1. (A138752(n) is the length of the so-called GB-sequence starting with prime(n).)
%H Paolo Xausa, <a href="/A138753/b138753.txt">Table of n, a(n) for n = 1..1459</a> (terms 1..500 from M. F. Hasler)
%H Georges Brougnard, <a href="https://www.echolalie.org/echolaliste/gbnums/gb0.jpg">Definition of GB-sequences</a>.
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F a(n) = min { k>0 | A138754^k(n) = A138754^m(n) for some m>=0, m<k }.
%F If n is not in {1,4,5,7}, then a(n) = 1+a(A138754(n)).
%e a(1)=1 since after 1 step we find 1 again.
%e a(4)=3 since 4 -> 7 -> 5 -> 4 under A138754.
%t A138754[n_]:=A138754[n]=With[{p=Prime[n]},PrimePi[NextPrime[If[Mod[p,3]==2,p/2,2p]]]];
%t A138753[n_]:=Length[NestWhileList[A138754,n,UnsameQ,{1,4}]]-1;
%t Array[A138753,100] (* _Paolo Xausa_, Jul 28 2023*)
%o (PARI) A138753(n,c=0,t=[1,1,1]) = { until( t[c++%3+1]==n=A138754(n), t[c%3+1]=n); c}
%Y Cf. A124123, A006577, A171938, A138756 (record values/indices).
%Y Cf. A138750, A138751, A138752, A138754.
%K nonn
%O 1,2
%A _M. F. Hasler_, Apr 01 2008