

A087256


Number of different initial values for 3x+1 trajectories in which the largest term appearing in the iteration is 2^n.


6



1, 1, 1, 6, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 13, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 11, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 21, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 78, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 9, 1, 3, 1
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OFFSET

1,4


COMMENTS

It would be interesting to know whether the ...1,3,1,3,1,x,1,3,1,3,1,... pattern persists.  John W. Layman, Jun 09, 2004
The observed pattern should persist. Proof: [1] a(odd)=1 because 1+2^odd is not divisible by 3, so in Collatzalgorithm 2^odd is preceded by increasing inverse step. Thus 2^odd is the only suitable initial value; [2] a[2k]>=3 for k>1 because 2^(2k)1=1+4^k=3A so {b=2^2k, (b1)/3 and (2a2)/3} are three relevant initial values. No more case arises unless condition[3] (see below) was satisfied; [3] a[6k+4]>=5 for k>=1, ..iv=c=2^(6k+4); here {c, (c1)/3, 2(c1)/3, (2c5)/9, (4c10)/9} is 5 suitable initial values, iff (2c5)/9 is integer; e.g. at 6k+4=10, {1024<341<682<227<454} backtracking the iteration.  Labos Elemer, Jun 17 2004
A105730 gives a(6k+4).  David Wasserman, Apr 18 2005


LINKS

Table of n, a(n) for n=1..103.


EXAMPLE

n = 10: 2^10 = 1024 = peak for trajectories started with initial value taken from the list: {151, 201, 227, 302, 341, 402, 454, 604, 682, 804, 908, 1024};
a trajectory with peak=1024: {201, 604, 302, 151, 454, 227, 682, 341, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}


MATHEMATICA

c[x_]:=c[x]=(1Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_]:=FixedPointList[c, x]; {$RecursionLimit=1000; m=0}; Table[Print[{xm1, m}]; m=0; Do[If[Equal[Max[fpl[n]], 2^xm], m=m+1], {n, 1, 2^xm}], {xm, 1, 30}]


PROG

(PARI) f(n, m) = 1 + if(2*n <= m, f(2*n, m), 0) + if (n%6 == 4, f(n\3, m), 0); a(n) = f(2^n, 2^n); (Wasserman)


CROSSREFS

Cf. A025586; A087251A087254.
Cf. A105730.
Sequence in context: A021066 A082730 A010136 * A154911 A152935 A102419
Adjacent sequences: A087253 A087254 A087255 * A087257 A087258 A087259


KEYWORD

nonn


AUTHOR

Labos Elemer, Sep 08 2003


EXTENSIONS

Terms a(19)a(21) from John W. Layman, Jun 09 2004
More terms from David Wasserman, Apr 18 2005


STATUS

approved



