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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). 4
 1, 3, 7, 15, 27, 27, 127, 255, 511, 1023, 1819, 4095, 4255, 16383, 32767, 65535, 77671, 262143, 459759, 1048575, 2097151, 4194303, 7456539, 16777215, 33554431, 67108863, 125687199, 125687199, 125687199, 1073741823, 2147483647, 4294967295, 8589934591, 17179869183 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The count of odd numbers includes the starting number n if it is part of the longest chain of odd numbers in the sequence. 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, ... 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 LINKS Kevin P. Thompson, Table of n, a(n) for n = 1..36 EXAMPLE 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. MAPLE nn:=200:T:=array(1..nn): for n from 1 to 20 do:jj:=0: for m from 3 by 2 to 10^8 while(jj=0) do: for i from 1 to nn while(jj=0) do: T[i]:=0:od:a:=1:T[1]:=m:x:=m: for it from 1 to 100 while (x>1) do: if irem(x, 2)=0 then x := x/2:a:=a+1:T[a]:=x: else x := 3*x+1: a := a+1: T[a]:=x: fi: od: jj:=0:aa:=a: for j from 1 to aa while(jj=0) do: if irem(T[j], 4)=3 then T[j]:=1: else T[j]:=0: fi: od: for p from 0 to aa-1 while (jj=0) do: s:=sum(T[p+k], k=1..2*n): if s=n then jj:=1: printf ( "%d %d \n", n, m): else fi: od: od: od: MATHEMATICA 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 *) CROSSREFS Cf. A006370, A006577, A000225, A024023, A213214. Cf. A222598 (similar). Sequence in context: A001649 A303220 A301894 * A353578 A324719 A170884 Adjacent sequences: A213212 A213213 A213214 * A213216 A213217 A213218 KEYWORD nonn AUTHOR Michel Lagneau, Mar 02 2013 EXTENSIONS Definition clarified, a(1) inserted, and a(21)-a(34) added by Kevin P. Thompson, Dec 15 2021 STATUS approved

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Last modified June 8 03:07 EDT 2023. Contains 363157 sequences. (Running on oeis4.)