A226672 -id:A226672 - cp's OEIS frontend

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

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.

A226688 Smallest positive integer in the primitive Collatz-like 3x-k cycle(s) with the conjectured record-breaking length A226686(n).

Original entry on oeis.org

17, 115, 2263, 113, 769, 436099, 6073, 3735185, 194197, 230891, 4575823, 823027, 21111517, 12477095

Offset: 1

Geoffrey H. Morley, Jun 16 2013

nonn

k = A226687(n).

Cf. A226672.

A226670 Record-breaking values, for increasing positive integers k == 1 or 5 mod 6, of the conjectured length of the longest primitive cycle(s) of positive integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

2, 27, 31, 43, 65, 66, 100, 106, 118, 136, 140, 141, 162, 200, 222, 262, 426, 476, 526, 636, 737, 1922, 2254, 4531, 4686, 5194, 5945, 9946, 10702, 14219, 16340, 19904, 37582, 40983, 49711, 63330

Offset: 1

Geoffrey H. Morley, Jun 16 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.

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).

k = A226671(n). The smallest integer in the T_k cycle(s) associated with a(n) is A226672(n).

Cf. A226609, A226686.

Definition clarified by Geoffrey H. Morley, Jun 23 2013

A226671 Integer k associated with the conjectured record-breaking length A226670(n) of primitive Collatz-like 3x+k cycles.

Original entry on oeis.org

1, 5, 17, 23, 29, 61, 85, 107, 125, 139, 143, 197, 253, 313, 371, 509, 563, 1135, 1163, 1307, 1699, 3299, 8431, 11491, 16819, 22097, 24917, 49787, 67475, 76733, 99391, 110273, 111611, 144379, 273641, 308219

Offset: 1

Geoffrey H. Morley, Jun 16 2013

nonn

Cf. A226672, A226687.

cp's OEIS Frontend

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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A226688 Smallest positive integer in the primitive Collatz-like 3x-k cycle(s) with the conjectured record-breaking length A226686(n).

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A226670 Record-breaking values, for increasing positive integers k == 1 or 5 mod 6, of the conjectured length of the longest primitive cycle(s) of positive integers under iteration by the Collatz-like 3x+k function.

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A226671 Integer k associated with the conjectured record-breaking length A226670(n) of primitive Collatz-like 3x+k cycles.

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