

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 p1 values {1, 2, 3, .., p1}.


1



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

1,1


COMMENTS

Or numbers n such that the multiplicative groups {n, T(n), T(T(n)),..., 4, 2, 1} / pZ are of order p1.
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 nonpowers 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,...


LINKS

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


EXAMPLE

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, 4, 5, 2, 1, 4, 2, 1} => the multiplicative group of order 6 is G = {1, 2, 3, 4, 5, 6}.


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=it1 and lst[1]=1
then
print(n):
else
fi:
fi:
od:


CROSSREFS

Cf. A001122, A006667, A214850.
Sequence in context: A017617 A246309 A038522 * A348093 A186293 A158865
Adjacent sequences: A267432 A267433 A267434 * A267436 A267437 A267438


KEYWORD

nonn


AUTHOR

Michel Lagneau, Jan 15 2016


EXTENSIONS

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


STATUS

approved



