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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}. 1
 8, 20, 32, 320, 2048, 2216, 8192, 13312, 87040, 218432, 524288, 89478400, 536870912, 137438953472, 250199979283796, 9007199254740992, 63800994005254144, 96076791692656640, 382805968326492160, 576460752303423488, 2305843009213693952, 4099276399740365440 (list; graph; refs; listen; history; text; internal format)
 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 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,... 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=it-1 and lst=1          then          print(n):          else         fi:       fi:     od: CROSSREFS Cf. A001122, A006667, A214850. Sequence in context: A017617 A246309 A038522 * A186293 A158865 A139570 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

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Last modified June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)