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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.

%I A226607 #37 Aug 28 2025 10:29:00
%S A226607 1,1,19,23,187,347,5,1,13,1,131,211,227,251,259,283,287,319,1,23,5,5,
%T A226607 7,41,7,17,1,11,3811,7055,13,13,17,19,23,29,1,1,5,25,65,73,85,89,101,
%U A226607 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 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.
%C A226607 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 A226607 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 A226607 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 A226607 Geoffrey H. Morley, <a href="/A226607/b226607.txt">Rows 1..6667 of array, flattened</a>
%H A226607 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 A226607 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 A226607 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 A226607 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 A226607 J. C. Lagarias, <a href="http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-aav56i1p33bwm">The set of rational cycles for the 3x+1 problem</a>, Acta Arith. 56 (1990), 33-53.
%e A226607 The irregular array starts:
%e A226607   (k=1)  1;
%e A226607   (k=5)  1, 19, 23, 187, 347;
%e A226607   (k=7)  5;
%e A226607   (k=11) 1, 13;
%e A226607 a(7)=5 is the smallest number in the primitive 3x+7 cycle {5,11,20,10}.
%Y A226607 Row n begins with a(A226612(n)) and has length A226613(n).
%Y A226607 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 A226607 Cf. A226623.
%K A226607 nonn,tabf,changed
%O A226607 1,3
%A A226607 _Geoffrey H. Morley_, Jun 13 2013
%E A226607 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.