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
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
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