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A267435 Numbers n such that each reduced Collatz trajectory (mod p): (n, T(n), T(T(n)),..., 4, 2, 1) / pZ, where the odd prime p is the number of iterations needed to reach 1, contains exactly the p-1 values {1, 2, 3, .., p-1}.

%I A267435 #17 Oct 17 2023 05:49:45
%S A267435 8,20,32,320,2048,2216,8192,13312,87040,218432,524288,89478400,
%T A267435 536870912,137438953472,250199979283796,9007199254740992,
%U A267435 63800994005254144,96076791692656640,382805968326492160,576460752303423488,2305843009213693952,4099276399740365440
%N A267435 Numbers n such that each reduced Collatz trajectory (mod p): (n, T(n), T(T(n)),..., 4, 2, 1) / pZ, where the odd prime p is the number of iterations needed to reach 1, contains exactly the p-1 values {1, 2, 3, .., p-1}.
%C A267435 Or numbers n such that the multiplicative groups {n, T(n), T(T(n)),..., 4, 2, 1} / pZ are of order p-1.
%C A267435 Property of the sequence:
%C A267435 This sequence provides a link with Artin’s conjecture on primitive roots.
%C A267435 Conjecture: the sequence is infinite (corollary of a Artin’s conjecture  because the sequence contains the numbers 2^A001122(k) where A001122 are the primes with primitive root 2).
%C A267435 The sequence is divided into two class of numbers:
%C A267435 i) A class of powers of 2: 2^3, 2^5, 2^11, 2^13, 2^19, 2^29, 2^37, 2^53, ..., 2^A001122(k),…
%C A267435 ii) A class of non-powers of 2: 20, 320, 2216, 13312, 87040, 218432, 89478400...
%C A267435 The corresponding p of the sequence are 3, 7, 5, 11, 11, 19, 13, 19, 19, 23, 19, 29,...
%H A267435 Hiroaki Yamanouchi, <a href="/A267435/b267435.txt">Table of n, a(n) for n = 1..37</a>
%H A267435 Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>.
%e A267435 20 is in the sequence because the Collatz trajectory of 20 is {20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1} with 7 iterations, and the corresponding reduced trajectory (mod 7) is {6, 3, 5, 2, 1, 4, 2, 1} => the multiplicative group of order 6 is G = {1, 2, 3, 4, 5, 6}.
%p A267435 nn:=10000:T:=array(1..2000):U:=array(1..2000):
%p A267435 for n from 1 to 10000000 do:
%p A267435   kk:=1:m:=n:T[kk]:=n:it:=0:
%p A267435     for i from 1 to nn while(m<>1) do:
%p A267435      if irem(m,2)=0
%p A267435        then
%p A267435        m:=m/2:kk:=kk+1:T[kk]:=m:it:=it+1:
%p A267435        else
%p A267435        m:=3*m+1:kk:=kk+1:T[kk]:=m:it:=it+1:
%p A267435      fi:
%p A267435     od:
%p A267435       if isprime(it)
%p A267435        then
%p A267435        lst:={}:
%p A267435        for p from 1 to it do:
%p A267435         lst:=lst union {irem(T[p],it)}:
%p A267435        od:
%p A267435         n0:=nops(lst):
%p A267435         if n0=it-1 and lst[1]=1
%p A267435          then
%p A267435          print(n):
%p A267435          else
%p A267435         fi:
%p A267435       fi:
%p A267435     od:
%Y A267435 Cf. A001122, A006667, A214850.
%K A267435 nonn
%O A267435 1,1
%A A267435 _Michel Lagneau_, Jan 15 2016
%E A267435 a(14)-a(22) from _Hiroaki Yamanouchi_, Jan 19 2016