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.

1, 1, 19, 23, 187, 347, 5, 1, 13, 1, 131, 211, 227, 251, 259, 283, 287, 319, 1, 23, 5, 5, 7, 41, 7, 17, 1, 11, 3811, 7055, 13, 13, 17, 19, 23, 29, 1, 1, 5, 25, 65, 73, 85, 89, 101, 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

Offset

1,3

Comments

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.

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.

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).

Links

Geoffrey H. Morley, Rows 1..6667 of array, flattened

E. G. Belaga and M. Mignotte, Cyclic Structure of Dynamical Systems Associated with 3x+d Extensions of Collatz Problem, Preprint math. 2000/17, Univ. Louis Pasteur, Strasbourg (2000).

E. G. Belaga and M. Mignotte, Walking Cautiously into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly, Fourth Colloquium on Mathematics and Computer Science, DMTCS proc. AG. (2006), 249-260.

E. G. Belaga and M. Mignotte, The Collatz Problem and Its Generalizations: Experimental Data. Table 1. Primitive Cycles of (3n+d)-mappings, Preprint math. 2006/15, Univ. Louis Pasteur, Strasbourg (2006).

E. G. Belaga and M. Mignotte, The Collatz Problem and Its Generalizations: Experimental Data. Table 2. Factorization of Collatz Numbers 2^l-3^k, Preprint math. 2006/15, Univ. Louis Pasteur, Strasbourg (2006).

J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arith. 56 (1990), 33-53.

Example

The irregular array starts:
(k=1)  1;
(k=5)  1, 19, 23, 187, 347;
(k=7)  5;
(k=11) 1, 13;
a(7)=5 is the smallest number in the primitive 3x+7 cycle {5,11,20,10}.

Crossrefs

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

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.

Cf. A226623.

Sequence in context: A033214 A107185 A240585 * A284495 A160077 A076353

Adjacent sequences: A226604 A226605 A226606 * A226608 A226609 A226610

Keyword

nonn,tabf

Author

Geoffrey H. Morley, Jun 13 2013

Extensions

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.

Status

approved