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A213215 For the Collatz (3x+1) iterations starting with the odd numbers k, a(n) is the smallest k such that the trajectory contains at least n successive odd numbers == 3 (mod 4).

%I A213215 #21 Dec 27 2021 10:36:57
%S A213215 1,3,7,15,27,27,127,255,511,1023,1819,4095,4255,16383,32767,65535,
%T A213215 77671,262143,459759,1048575,2097151,4194303,7456539,16777215,
%U A213215 33554431,67108863,125687199,125687199,125687199,1073741823,2147483647,4294967295,8589934591,17179869183
%N A213215 For the Collatz (3x+1) iterations starting with the odd numbers k, a(n) is the smallest k such that the trajectory contains at least n successive odd numbers == 3 (mod 4).
%C A213215 The count of odd numbers includes the starting number n if it is part of the longest chain of odd numbers in the sequence.
%C A213215 The sequence is infinite because the Collatz trajectory starting at k = 2^n - 1 contains at least n consecutive odd numbers == 3 (mod 4) such that 3*2^n - 1 -> 3^2*2^(n-1)-1 -> ... -> 2*3^(n-1)-1 and then -> 3^n-1 -> ...  but the numbers of this sequence are not always of this form, for example 27, 1819, 4255, 77671, 459759, ...
%C A213215 Equivalently, a(n) is the smallest k such that the Collatz sequence for k suffers at least n consecutive (3x+1)/2 operations (i.e., no consecutive divisions by 2). - _Kevin P. Thompson_, Dec 15 2021
%H A213215 Kevin P. Thompson, <a href="/A213215/b213215.txt">Table of n, a(n) for n = 1..36</a>
%e A213215 a(4)=15 because the Collatz sequence for 15 (15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1) is the first Collatz sequence to contain 4 consecutive odd numbers congruent to 3 (mod 4): 15, 23, 35, and 53.
%p A213215 nn:=200:T:=array(1..nn):
%p A213215 for n from 1 to 20 do:jj:=0:
%p A213215          for m from 3 by 2 to 10^8 while(jj=0) do:
%p A213215                for i from 1 to nn while(jj=0) do:
%p A213215                T[i]:=0:od:a:=1:T[1]:=m:x:=m:
%p A213215                      for it from 1 to 100 while (x>1) do:
%p A213215                          if irem(x,2)=0 then
%p A213215                          x := x/2:a:=a+1:T[a]:=x:
%p A213215                          else
%p A213215                          x := 3*x+1: a := a+1: T[a]:=x:
%p A213215                         fi:
%p A213215                      od:
%p A213215                      jj:=0:aa:=a:
%p A213215                        for j from 1 to aa while(jj=0) do:
%p A213215                          if irem(T[j],4)=3 then
%p A213215                          T[j]:=1:
%p A213215                          else
%p A213215                          T[j]:=0:
%p A213215                        fi:
%p A213215                       od:
%p A213215                          for p from 0 to aa-1 while (jj=0) do:
%p A213215                          s:=sum(T[p+k],k=1..2*n):
%p A213215                          if s=n then
%p A213215                          jj:=1: printf ( "%d %d \n",n,m):
%p A213215                          else
%p A213215                          fi:
%p A213215                   od:
%p A213215               od:
%p A213215            od:
%t A213215 Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countThrees[t_] := Module[{mx = 0, cnt = 0, i = 0}, While[i < Length[t], i++; If[t[[i]] == 3, cnt++; i++, If[cnt > mx, mx = cnt]; cnt = 0]]; mx]; nn = 15; t = Table[0, {nn}]; n = 1; While[Min[t] == 0, n = n + 2; c = countThrees[Mod[Collatz[n], 4]]; If[c <= nn && t[[c]] == 0, t[[c]] = n; Do[If[t[[i]] == 0, t[[i]] = n], {i, c}]]]; t (* _T. D. Noe_, Mar 02 2013 *)
%Y A213215 Cf. A006370, A006577, A000225, A024023, A213214.
%Y A213215 Cf. A222598 (similar).
%K A213215 nonn
%O A213215 1,2
%A A213215 _Michel Lagneau_, Mar 02 2013
%E A213215 Definition clarified, a(1) inserted, and a(21)-a(34) added by _Kevin P. Thompson_, Dec 15 2021