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A214850 3x+1 groups: irregular triangle read by rows: row n gives all the numbers p <= A075684(n)+1 such that {T(2n+1,k) /pZ} is a multiplicative finite group, where T(2n+1,k) is the reduced trajectory of the Collatz problem whose elements are all odd.

%I A214850 #43 Apr 02 2017 00:23:26
%S A214850 2,4,2,4,6,2,4,6,8,12,18,2,4,8,2,4,6,2,4,6,8,12,2,4,8,2,4,6,8,12,2,4,
%T A214850 6,12,2,4,8,10,20,22,2,4,6,8,2,4,6,12,18,2,4,8,16,2,4,6,2,4,6,8,12,16,
%U A214850 18,24,2,4,2,4,6,2,4,6,8,12,18,2,4,8,2,4,6
%N A214850 3x+1 groups: irregular triangle read by rows: row n gives all the numbers p <= A075684(n)+1 such that {T(2n+1,k) /pZ} is a multiplicative finite group, where T(2n+1,k) is the reduced trajectory of the Collatz problem whose elements are all odd.
%C A214850 We introduce the structure of a finite group in order to give a possible way to classify the Collatz trajectories.
%C A214850 We see that the study of the classification of the trajectories is dependent on the values p.
%C A214850 The principle of the algorithm is to compute all the products T(2n+1,i)/pZ * T(2n+1,j)/pZ and also the inverse of each element such that if x is in the group, then there exist x' in the group with x*x' = 1.
%C A214850 Rows of triangle:
%C A214850 {2, 4},
%C A214850 {2, 4, 6},
%C A214850 {2, 4, 6, 8, 12, 18},
%C A214850 {2, 4, 8},
%C A214850 {2, 4, 6},
%C A214850 {2, 4, 6, 8, 12},
%C A214850 {2, 4, 8},
%C A214850 {2, 4, 6, 8, 12}, ...
%H A214850 Michel Lagneau, <a href="/A214850/b214850.txt">Rows n = 1..400 of irregular triangle, flattened</a>
%e A214850 Row 18 gives 6 groups with p = {2, 4, 6, 8, 12, 18}. The Collatz trajectory T(37,k) = {37, 7, 11, 17, 13, 5, 1}, and if we choose, for example, p=18, we obtain G(37) = {T(37,k)/18Z} = {7, 11, 17, 13, 5, 1} which (as subset of Z/18Z) is a multiplicative group of order 6.
%e A214850 For example 5, or 11, generates the cyclic group:
%e A214850 5^1 == 5, 5^2 == 7, 5^3 == 17, 5^4 == 13, 5^5 == 11, 5^6 == 1 (mod 18).
%e A214850 Other subgroups are {1}, {1, 17} and {1, 7, 13}.
%p A214850 c:=0:
%p A214850 for n from 3 by 2 to 800 do:
%p A214850        x:=2:lst:={n}:lst1:={}:x := n:
%p A214850           for k from 1 to 120 while (x > 1) do:
%p A214850              a := 0:
%p A214850              if type(x, 'even') then
%p A214850              x := x/2:lst:=lst union {x}:a:=a+1:
%p A214850             else
%p A214850             x := 3*x+1 :lst:=lst union {x}:a:=a+1:
%p A214850             fi:
%p A214850        od:
%p A214850         n1:=nops(lst):
%p A214850           for u from 1 to n1 do:
%p A214850              if irem(lst[u], 2)=1 then
%p A214850              lst1:=lst1 union {lst[u]}:
%p A214850             else
%p A214850            fi:
%p A214850          od:
%p A214850          m1:= max( op(lst1)):n1:=nops(lst1):
%p A214850             for p from 2 by 2 to m1+1 do:
%p A214850             lst2:={}:
%p A214850                for q from 1 to n1 do:
%p A214850                lst2:=lst2 union {irem(lst1[q], p)}:
%p A214850               od:
%p A214850               lst3:={}:n2:=nops(lst2):kkk:=0:
%p A214850                 for i from 1 to n2 do:jjj:=0:
%p A214850                     for j from 1 to n2 do:
%p A214850                        z:=irem(lst2[i]*lst2[j], p):lst3:=lst3 union{z}:
%p A214850                        if z=1 then jjj:=1:else fi
%p A214850                     od:
%p A214850                     if jjj=0 then kkk:=1:else fi:
%p A214850                  od:
%p A214850                  n3:=nops(lst3):iii:=0:
%p A214850                      for b from 1 to n3 while(iii=0 and n2=n3 and kkk=0)
%p A214850                  do:
%p A214850                       if lst2[b]<>lst3[b] then
%p A214850                       iii:=1:else
%p A214850                       fi:
%p A214850                     od:
%p A214850                     if iii=0  and n2=n3 and kkk=0 then c:=c+1:
%p A214850                     printf ( "%d %d \n",c,p):
%p A214850                     else
%p A214850                     fi:
%p A214850                 od:
%p A214850                  x:=2:
%p A214850               od:
%Y A214850 Cf. A075680, A075684.
%K A214850 nonn,tabf
%O A214850 1,1
%A A214850 _Michel Lagneau_, Mar 08 2013