A226610 -id:A226610 - cp's OEIS frontend

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

Geoffrey H. Morley, Jun 13 2013

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).

			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}.

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.

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.

For 0

A226609 Irregular array read by rows. a(n) is the length of the primitive Collatz-like 3x+k cycle associated with A226607(n).

Original entry on oeis.org

2, 3, 5, 5, 27, 27, 4, 6, 14, 4, 24, 8, 8, 8, 8, 8, 8, 8, 7, 31, 11, 5, 5, 43, 16, 8, 5, 17, 65, 65, 23, 8, 8, 6, 6, 6, 20, 11, 18, 28, 7, 7, 7, 7, 7, 38, 29, 12, 6, 28, 28, 10, 10, 10, 10, 10, 10, 6, 66, 24, 30, 10, 10, 27, 27, 27, 27, 27, 12, 60, 15, 38

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  2;
(k=5)  3, 5, 5, 27, 27;
(k=7)  4;
(k=11) 6, 14;
a(2)=3 is the length of the 3x+5 cycle {1,4,2} associated with A226607(2)=1.

Geoffrey H. Morley, Rows 1..2032 of array, flattened

Row n begins with a(A226612(n)) and has length A226613(n).
The cycle associated with a(n) has A226610(n) odd elements of which A226608(n) is the largest.
Cf. A226611, A226625, A226670-A226672.

A226626 Irregular array read by rows. a(n) is the number of odd elements in the primitive Collatz-like 3x-k cycle associated with A226623(n).

Original entry on oeis.org

1, 2, 7, 3, 4, 4, 11, 12, 12, 12, 12, 12, 12, 12, 12, 22, 8, 6, 4, 14, 14, 14, 8, 11, 11, 11, 44, 5, 5, 5, 12, 28, 14, 24, 24, 24, 24, 24, 14, 14, 21, 21, 14, 7, 7, 14, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 32, 8

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1, 2, 7;
(k=11) 3;
(k=17) 4, 4;
(k=19) 11;
a(4)=3 is the number of odd elements in the 3x-11 cycle {19,23,29,38} associated with A226623(4)=19

Geoffrey H. Morley, Rows 1..280 of array, flattened

Row n begins with a(A226628(n)) and has length A226629(n). k=A226630(n).
The cycle associated with a(n) has length A226625(n) and its largest element is A226624(n).
Cf. A226610, A226627, A226632, A226689.

A226608 Irregular array read by rows. a(n) is the largest element in the primitive Collatz-like 3x+k cycle associated with A226607(n).

Original entry on oeis.org

1, 1, 49, 37, 2773, 3397, 11, 7, 79, 1, 1853, 1121, 797, 665, 905, 653, 761, 557, 5, 181, 35, 19, 11, 1651, 137, 41, 1, 121, 2277097, 1051393, 131, 127, 79, 89, 53, 65, 157, 23, 43, 643, 331, 223, 211, 259, 175, 1409, 757, 71, 19, 827, 139, 1399, 775, 751, 967, 559, 571

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1;
(k=5)  1, 49, 37, 2772, 3397;
(k=7)  11;
(k=11) 7, 79;
a(3)=49 is the largest element in the 3x+5 cycle {19,31,49,76,38} associated with A226607(3)=19.

Geoffrey H. Morley, Rows 1..2032 of array, flattened

Row n begins with a(A226612(n)) and has length A226613(n).
The cycle associated with a(n) has length A226609(n) and A226610(n) odd elements.
Cf. A226611, A226624, A226667-A226669.

A226611 Irregular array read by rows. a(n) is the smallest starting value of a T_k trajectory that includes A226607(n), where T_k is the 3x+k function associated with A226607(n).

Original entry on oeis.org

1, 1, 3, 23, 123, 171, 1, 1, 3, 1, 19, 99, 147, 163, 123, 283, 159, 319, 1, 9, 1, 5, 7, 1, 1, 3, 1, 3, 2531, 5859, 1, 1, 3, 1, 3, 7, 1, 1, 5, 1, 9, 33, 39, 21, 101, 1, 1, 1, 7, 9, 1, 3, 149, 21, 93, 125, 221, 1, 175, 1, 1, 1, 7, 2585, 1073, 2301, 4121, 893

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1;
(k=5)  1, 3, 23, 123, 171;
(k=7)  1;
(k=11) 1, 3;
a(3)=3 is the smallest starting value for a 3x+5 trajectory that includes A226607(3)=19. The trajectory is {3,7,13,22,11,19,...}.

Geoffrey H. Morley, Rows 1..2032 of array, flattened

Row n begins with a(A226612(n)) and has length A226613(n).
The cycle associated with a(n) has length A226609(n) and A226610(n) odd elements of which A226608(n) is the largest.
Cf. A226627.

A226661 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, exactly n of which are odd, under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 7, 5, 25, 13, 59, 47, 11, 29, 145, 59, 31, 115, 79, 13, 47, 5, 17, 125, 79, 263, 49, 169, 91, 191, 23, 601, 323, 193, 109, 311, 73, 149, 265, 571, 95, 491, 697, 695, 137, 29, 119, 383, 575, 283, 121, 263, 233, 163, 193, 283, 479, 107, 203, 437, 85, 491, 349

Offset: 1

Geoffrey H. Morley, Jul 05 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
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, where k is odd.
For primitive cycles, GCD(k,6)=1.
Conjecture: a(n)>0 for all n.

Geoffrey H. Morley, Table of n, a(n) for n = 1..2520

Cf. A226610, A226660, A226677.

A226673 Conjectured record-breaking numbers of odd elements, for ascending positive integers k, in primitive cycles of positive integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 17, 18, 26, 41, 56, 74, 80, 89, 115, 126, 142, 215, 220, 256, 311, 387, 1000, 1136, 1146, 2253, 2292, 2590, 2937, 4971, 5326, 7157, 8294, 9920, 18862, 20429, 24842, 31913

Offset: 1

Geoffrey H. Morley, Jun 19 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
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.
For primitive cycles, GCD(k,6)=1.

k = A226674(n). The smallest integer in the T_k cycle associated with a(n) is A226675(n).

Cf. A226610, A226689.

A226617 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, n of which are odd including 1, under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 11, 43, 55, 643, 97, 673, 41, 1843, 329, 59, 113, 5603, 289, 6505, 77, 407, 127, 499, 79, 865, 749

Offset: 1

Geoffrey H. Morley, Jul 03 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
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, where k is odd.
For primitive cycles, GCD(k,6)=1.
Conjecture: a(n)>0 for all n.

			The cycle associated with a(1)=1 is {1,2}, with a(2)=11 is {1,7,16,8,4,2}, and with a(3)=43 is {1,23,56,28,14,7,32,16,8,4,2}.

Cf. A226607, A226610, A226616.

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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.

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A226609 Irregular array read by rows. a(n) is the length of the primitive Collatz-like 3x+k cycle associated with A226607(n).

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A226626 Irregular array read by rows. a(n) is the number of odd elements in the primitive Collatz-like 3x-k cycle associated with A226623(n).

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A226608 Irregular array read by rows. a(n) is the largest element in the primitive Collatz-like 3x+k cycle associated with A226607(n).

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A226611 Irregular array read by rows. a(n) is the smallest starting value of a T_k trajectory that includes A226607(n), where T_k is the 3x+k function associated with A226607(n).

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A226661 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, exactly n of which are odd, under iteration by the Collatz-like 3x+k function.

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A226673 Conjectured record-breaking numbers of odd elements, for ascending positive integers k, in primitive cycles of positive integers under iteration by the Collatz-like 3x+k function.

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A226617 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, n of which are odd including 1, under iteration by the Collatz-like 3x+k function.

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