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Smallest number which requires n^2 steps in the 3x+1 problem.
2

%I #14 Apr 24 2022 17:19:16

%S 1,2,16,12,7,98,153,169,673,350,107,129,649,2110,4763,6919,6943,

%T 158299,71310,724686,845223,665215,2157291,4468260,5978623,34385063,

%U 21015006,301695657,489853918,568097511,418606034,1208474114

%N Smallest number which requires n^2 steps in the 3x+1 problem.

%C The next term is >= 1410123943. - Larry Reeves (larryr(AT)acm.org), Apr 10 2002

%H Jeffrey C. Lagarias, <a href="http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html">The 3x + 1 Problem and its Generalizations</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%e a(3)=12 since the Collatz sequence is 12 -> 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 and the count of steps is 9.

%t f[ n_ ] := Module[ {i = 0, m = n}, While[ m != 1, m = If[ OddQ[ m ], 3m + 1, m/2 ]; i++ ]; i ]; a = Table[ 0, {50} ]; Do[ m = Sqrt[ f[ n ] ]; If[ IntegerQ[ m ] && a[ [ m + 1 ] ] == 0, a[ [ m + 1 ] ] = n ] ], {n, 1, 10^6} ]; a

%o (PARI) {sequence(n)= c=0; k=n; while(k>1, if(k%2==0,k=k/2,k=3*k+1); c=c+1; ); if(issquare(c),print(n," ",c),); }

%Y Cf. A006577, A066905.

%K nonn

%O 0,2

%A _Randall L Rathbun_, Jan 17 2002

%E Corrected and extended by _Dean Hickerson_ and _Robert G. Wilson v_, Jan 18 2002

%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 10 2002