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A135282 Largest k such that 2^k appears in the trajectory of the Collatz 3x+1 sequence started at n. 13
0, 1, 4, 2, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 6, 8, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Most of the first eighty terms in the sequence are 4, because the trajectories finish with 16 -> 8 -> 4 -> 2 -> 1. - R. J. Mathar, Dec 12 2007

Most of the first ten thousand terms are 4, and there only appear 4, 6, 8, and 10 in the extent, unless n is power of 2. In the other words, it seems that the trajectory of the Collatz 3x + 1 sequence ends with either 16, 64, 256 or 1024. There are few exceptional terms, for example a(10920) = 12, a(10922) = 14. It also seems all terms are even unless n is an odd power of 2. - Masahiko Shin, Mar 16 2010

It is true that all terms are even unless n is an odd power of 2: 2 == -1 mod 3, 2 * 2 == -1 * -1 == 1 mod 3. Therefore only even-indexed powers of 2 are congruent to 1 mod 3 and thus reachable by either a halving step or a "tripling step," whereas the odd-indexed powers of 2 are only reachable by a halving step or as an initial value. - Alonso del Arte, Aug 15 2010

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Index entries related to the 3x+1 (Collatz) problem.

FORMULA

a(n) = A006577(n) - A208981(n) (after Alonso del Arte's comment in A208981), if A006577(n) is not -1. - Omar E. Pol, Apr 10 2022

EXAMPLE

a(6) = 4 because the sequence is 6, 3, 10, 5, 16, 8, 4, 2, 1; there 16 = 2^4 is the largest power of 2 encountered.

MAPLE

A135282 := proc(n) local k, threen1 ; k := 0 : threen1 := n ; while threen1 > 1 do if 2^ilog[2](threen1) = threen1 then k := max(k, ilog[2](threen1)) ; fi ; if threen1 mod 2 = 0 then threen1 := threen1/2 ; else threen1 := 3*threen1+1 ; fi ; od: RETURN(k) ; end: for n from 1 to 80 do printf("%d, ", A135282(n)) ; od: # R. J. Mathar, Dec 12 2007

MATHEMATICA

Collatz[n_] := If[EvenQ[n], n/2, 3*n + 1]; Log[2, Table[NestWhile[Collatz, n, ! IntegerQ[Log[2, #]] &], {n, 100}]] (* T. D. Noe, Mar 05 2012 *)

PROG

(C) #include <stdio.h> int main(){ int i, s, f; for(i = 2; i < 10000; i++){ f = 0; s = i; while(s != 1){ if(s % 2 == 0){ s = s/2; f++; } else{ f = 0; s = 3 * s + 1; } } printf("%d, ", f); } return 0; } /* Masahiko Shin, Mar 16 2010 */

(Haskell)

a135282 = a007814 . head . filter ((== 1) . a209229) . a070165_row

-- Reinhard Zumkeller, Jan 02 2013

CROSSREFS

Cf. A007814, A209229, A070165, A232503.

Cf. A006577, A208981.

Sequence in context: A232715 A317951 A095382 * A347409 A179411 A103859

Adjacent sequences: A135279 A135280 A135281 * A135283 A135284 A135285

KEYWORD

nonn

AUTHOR

Masahiko Shin, Dec 02 2007

EXTENSIONS

Edited and extended by R. J. Mathar, Dec 12 2007

More terms from Masahiko Shin, Mar 16 2010

STATUS

approved

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Last modified December 7 05:07 EST 2022. Contains 358649 sequences. (Running on oeis4.)