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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 n successive odd numbers == 3 (mod 4). 1

%I

%S 3,7,15,27,27,127,255,511,1023,1819,4095,4255,16383,32767,65535,77671,

%T 262143,459759,1048575

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

%C n is possibly counted.

%C 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, ...

%e a(3)=15 because the Collatz iterations starting at 15 and 6 are : 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 and the 3 consecutive odd numbers 15, 23 and 35 are congruent to 3 (mod 4).

%p nn:=200:T:=array(1..nn):

%p for n from 1 to 20 do:jj:=0:

%p for m from 3 by 2 to 10^8 while(jj=0) do:

%p for i from 1 to nn while(jj=0) do:

%p T[i]:=0:od:a:=1:T[1]:=m:x:=m:

%p for it from 1 to 100 while (x>1) do:

%p if irem(x,2)=0 then

%p x := x/2:a:=a+1:T[a]:=x:

%p else

%p x := 3*x+1: a := a+1: T[a]:=x:

%p fi:

%p od:

%p jj:=0:aa:=a:

%p for j from 1 to aa while(jj=0) do:

%p if irem(T[j],4)=3 then

%p T[j]:=1:

%p else

%p T[j]:=0:

%p fi:

%p od:

%p for p from 0 to aa-1 while (jj=0) do:

%p s:=sum(T[p+k],k=1..2*n):

%p if s=n then

%p jj:=1: printf ( "%d %d \n",n,m):

%p else

%p fi:

%p od:

%p od:

%p od:

%t 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 Cf. A006577, A000225, A024023, A213214.

%Y Cf. A222598 (similar).

%K nonn

%O 1,1

%A _Michel Lagneau_, Mar 02 2013

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Last modified June 13 04:04 EDT 2021. Contains 344980 sequences. (Running on oeis4.)