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A226607 Irregular array read by rows in which row floor(k/3)+1, where GCD(k,6)=1, lists the smallest elements, in ascending order, of conjecturally all primitive cycles of positive integers under iteration by the 3x+k function. 12

%I

%S 1,1,19,23,187,347,5,1,13,1,131,211,227,251,259,283,287,319,1,23,5,5,

%T 7,41,7,17,1,11,3811,7055,13,13,17,19,23,29,1,1,5,25,65,73,85,89,101,

%U 25,103,1,7,41,1,133,149,181,185,217,221,1,235,19,17,29,31,2585,2809,3985,4121,4409,5,19,47,1,1,7,233,265

%N Irregular array read by rows in which row floor(k/3)+1, where GCD(k,6)=1, lists the smallest elements, in ascending order, of conjecturally all primitive cycles of positive integers under iteration by the 3x+k function.

%C The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd.

%C Lagarias (1990) called a T_k cycle primitive if its elements are all relatively prime to k or, equivalently, if its elements are not a common multiple of the elements of another cycle. He conjectured that, for every positive integer k relatively prime to 6, there is at least one primitive cycle of the map T_k and that the number of such cycles is finite.

%C For k<158195 no trajectory with a starting value below 10^8 has a primitive cycle whose minimal element exceeds 28306063 (attained when k=103645). This suggests that the 42757 primitive cycles found for k<20000, by examining every trajectory with a starting value below 10^8, are complete. Their largest minimal element is 8013899 (when k=17021).

%H Geoffrey H. Morley, <a href="/A226607/b226607.txt">Rows 1..6667 of array, flattened</a>

%H E. G. Belaga and M. Mignotte, <a href="http://hal.archives-ouvertes.fr/hal-00129656">Cyclic Structure of Dynamical Systems Associated with 3x+d Extensions of Collatz Problem</a>, Preprint math. 2000/17, Univ. Louis Pasteur, Strasbourg (2000).

%H E. G. Belaga and M. Mignotte, <a href="http://hal.archives-ouvertes.fr/hal-00129726">Walking Cautiously into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly</a>, Fourth Colloquium on Mathematics and Computer Science, DMTCS proc. AG. (2006), 249-260.

%H E. G. Belaga and M. Mignotte, <a href="http://hal.archives-ouvertes.fr/hal-00129727">The Collatz Problem and Its Generalizations: Experimental Data. Table 1. Primitive Cycles of (3n+d)-mappings</a>, Preprint math. 2006/15, Univ. Louis Pasteur, Strasbourg (2006).

%H E. G. Belaga and M. Mignotte, <a href="http://hal.archives-ouvertes.fr/hal-00129730">The Collatz Problem and Its Generalizations: Experimental Data. Table 2. Factorization of Collatz Numbers 2^l-3^k</a>, Preprint math. 2006/15, Univ. Louis Pasteur, Strasbourg (2006).

%H J. C. Lagarias, <a href="http://pldml.icm.edu.pl:80/mathbwn/element/bwmeta1.element.bwnjournal-article-aav56i1p33bwm?q=bwmeta1.element.bwnjournal-number-aa-1990-56-1&amp;qt=CHILDREN-STATELESS">The set of rational cycles for the 3x+1 problem,</a> Acta Arith. 56 (1990), 33-53.

%e The irregular array starts:

%e (k=1) 1;

%e (k=5) 1, 19, 23, 187, 347;

%e (k=7) 5;

%e (k=11) 1, 13;

%e a(7)=5 is the smallest number in the primitive 3x+7 cycle {5,11,20,10}.

%Y Row n begins with a(A226612(n)) and has length A226613(n).

%Y The smallest starting value whose trajectory includes a(n) is A226611(n). The cycle associated with a(n) has length A226609(n) and A226610(n) odd elements of which A226608(n) is the largest.

%Y Cf. A226623.

%K nonn,tabf

%O 1,3

%A _Geoffrey H. Morley_, Jun 13 2013

%E For 0<k<20000 Belaga and Mignotte (2000)'s Table 3 overcounts the d's (our k's) with both 3 and 5 cycles by 1, making their count of 42765 known cycles excessive by 8.

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Last modified March 26 16:33 EDT 2019. Contains 321510 sequences. (Running on oeis4.)