

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 evenindexed powers of 2 are congruent to 1 mod 3 and thus reachable by either a halving step or a "tripling step," whereas the oddindexed 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



