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%I #23 Sep 23 2023 03:13:51
%S 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,
%T 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,
%U 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
%N Number of different initial values for 3x+1 trajectories in which the largest term appearing in the iteration is 2^n.
%C 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
%C The observed pattern should persist. Proof: [1] a(odd)=1 because -1+2^odd is not divisible by 3, so in Collatz-algorithm 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, (b-1)/3 and (2a-2)/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, (c-1)/3, 2(c-1)/3, (2c-5)/9, (4c-10)/9} is 5 suitable initial values, iff (2c-5)/9 is integer; e.g. at 6k+4=10, {1024<-341<-682<-227<-454} back-tracking the iteration. - _Labos Elemer_, Jun 17 2004
%C From _Hartmut F. W. Hoft_, Jun 24 2016: (Start)
%C Except for a(2)=1 the sequence has the 6-element 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 (2n-5)/9 is an integer and a(n)>=6.
%C Conjecture: All numbers m > 10 occur as values in A087256 (see A233293).
%C 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.
%C (End)
%H Hartmut F. W. Hoft, <a href="/A087256/a087256.pdf">proof of quasi period 6</a>
%F a(6n+4) = A105730(n). - _David Wasserman_, Apr 18 2005
%e 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};
%e a trajectory with peak=1024: {201, 604, 302, 151, 454, 227, 682, 341, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}
%t c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=FixedPointList[c, x]; {$RecursionLimit=1000;m=0}; Table[Print[{xm-1, m}];m=0; Do[If[Equal[Max[fpl[n]], 2^xm], m=m+1], {n, 1, 2^xm}], {xm, 1, 30}]
%o (PARI) f(n, m) = 1 + if(2*n <= m, f(2*n, m), 0) + if (n%6 == 4, f(n\3, m), 0);
%o a(n) = f(2^n, 2^n); \\ _David Wasserman_, Apr 18 2005
%Y Cf. A025586, A087251, A087252, A087253, A087254, A105730, A233293.
%K nonn
%O 1,4
%A _Labos Elemer_, Sep 08 2003
%E Terms a(19)-a(21) from _John W. Layman_, Jun 09 2004
%E More terms from _David Wasserman_, Apr 18 2005
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