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}.

8, 20, 32, 320, 2048, 2216, 8192, 13312, 87040, 218432, 524288, 89478400, 536870912, 137438953472, 250199979283796, 9007199254740992, 63800994005254144, 96076791692656640, 382805968326492160, 576460752303423488, 2305843009213693952, 4099276399740365440

Offset: 1

Author

Michel Lagneau, Jan 15 2016

Keywords

• nonn

Comments

Or numbers n such that the multiplicative groups {n, T(n), T(T(n)),..., 4, 2, 1} / pZ are of order p-1.

Property of the sequence:

This sequence provides a link with Artin’s conjecture on primitive roots.

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).

The sequence is divided into two class of numbers:

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),…

ii) A class of non-powers of 2: 20, 320, 2216, 13312, 87040, 218432, 89478400...

The corresponding p of the sequence are 3, 7, 5, 11, 11, 19, 13, 19, 19, 23, 19, 29,...

Examples

			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}.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..37
• Wikipedia, Artin's conjecture on primitive roots.

Crossrefs

Cf. A001122, A006667, A214850.

Programs

Maple

nn:=10000:T:=array(1..2000):U:=array(1..2000):
for n from 1 to 10000000 do:
  kk:=1:m:=n:T[kk]:=n:it:=0:
    for i from 1 to nn while(m<>1) do:
     if irem(m,2)=0
       then
       m:=m/2:kk:=kk+1:T[kk]:=m:it:=it+1:
       else
       m:=3*m+1:kk:=kk+1:T[kk]:=m:it:=it+1:
     fi:
    od:
      if isprime(it)
       then
       lst:={}:
       for p from 1 to it do:
        lst:=lst union {irem(T[p],it)}:
       od:
        n0:=nops(lst):
        if n0=it-1 and lst[1]=1
         then
         print(n):
         else
        fi:
      fi:
    od:

Extensions

a(14)-a(22) from Hiroaki Yamanouchi, Jan 19 2016