

A087256


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


7



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
From Hartmut F. W. Hoft, Jun 24 2016: (Start)
Except for a(2)=1 the sequence has the 6element quasiperiod 1, 3, 1, x, 1, 3 where x>=6, but unequal to 7 and 10 (see links below and in A033496). Observe that for n=2^(6k+4)=16*2^(6k), n mod 9 = 7 so that (2n5)/9 is an integer and a(n)>=6.
Conjecture: All numbers m > 10 occur as values in A087256 (see A233293).
The conjecture has been verified for all 10 < k < 133 for Collatz trajectories with maximum value through 2^(36000*6 + 4). The largest fan of initial values in this range, F(6*1993+4), has maximum 2^11962 and size 3958.
(End)


LINKS

Table of n, a(n) for n=1..103.
Hartmut F. W. Hoft, proof of quasi period 6


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); \\ David Wasserman


CROSSREFS

Cf. A025586, A087251A087254, A105730, A233293.
Sequence in context: A021066 A082730 A010136 * A154911 A152935 A253686
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



