A226609 -id:A226609 - 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

A226610 Irregular array read by rows. a(n) is the number of odd elements in the primitive 3x+k cycle associated with A226607(n).

Original entry on oeis.org

1, 1, 3, 3, 17, 17, 2, 2, 8, 1, 15, 5, 5, 5, 5, 5, 5, 5, 2, 18, 5, 2, 2, 26, 8, 4, 1, 9, 41, 41, 12, 4, 4, 3, 3, 3, 8, 3, 7, 16, 4, 4, 4, 4, 4, 22, 17, 4, 2, 16, 11, 6, 6, 6, 6, 6, 6, 1, 41, 12, 16, 5, 5, 17, 17, 17, 17, 17, 4, 32, 8, 16, 20, 20, 14, 14

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1;
(k=5)  1, 3, 3, 17, 17;
(k=7)  2;
(k=11) 2, 8;
a(2)=1 is the number of odd elements in 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 length A226609(n) and its largest element is A226608(n).
Cf. A226611, A226626.

A226625 Irregular array read by rows. a(n) is the length of the primitive Collatz-like 3x-k cycle associated with A226623(n).

Original entry on oeis.org

1, 3, 11, 4, 6, 6, 17, 19, 19, 19, 19, 19, 19, 19, 19, 34, 12, 9, 5, 22, 22, 22, 12, 17, 17, 17, 69, 7, 7, 7, 18, 44, 22, 38, 38, 38, 38, 38, 22, 22, 33, 33, 22, 11, 11, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 48, 12

Offset: 1

Geoffrey H. Morley, Jun 13 2013

Conjecture: Every cycle with the same value of k (k>1) has the same proportion of odd and even elements. Thus if n>1 then A226626(n)/A226625(n) has the same value for each m where A226628(n) <= m < A226628(n+1).

			The irregular array starts:
(k=1)  1, 3, 11;
(k=11) 4;
(k=17) 6, 6;
(k=19) 17;
a(4)=4 is the length of 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 A226626(n) odd elements of which A226624(n) is the largest.
Cf. A226609, A226627, A226631, A226686, A226687, A226688.

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.

A226618 Irregular array read by rows in which row n lists the positive integers k in ascending order for which 1 is in a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 13, 29, 11, 61, 17, 125, 253, 509, 145, 203, 1021, 43, 2045, 55, 4093, 355, 1169, 8189, 137, 3275, 16381, 1129, 32765, 1007, 5957, 9361, 65533, 131069, 97, 52427, 262141, 643, 74897, 524285, 41, 1048573, 553, 28727, 110375, 2097149, 281, 673, 2075, 9731, 34663

Offset: 1

Geoffrey H. Morley, Jul 02 2013

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.

			The irregular array starts:
1;
5;
13;
29;
11, 61;
17, 125; ...
Row 1 is empty.

Geoffrey H. Morley, Rows 2..26 of array, flattened

The first element in row n is A226616(n), and the last is A036563(n) = 2^n-3.

Cf. A226609, A226619.

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

A226660 Smallest positive integer k with a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 7, 5, 11, 17, 13, 97, 59, 19, 55, 233, 11, 73, 25, 29, 47, 215, 41, 103, 145, 31, 13, 119, 131, 5, 47, 53, 67, 17, 337, 125, 115, 485, 133, 127, 49, 119, 191, 293, 133, 23, 79, 103, 191, 167, 91, 409, 329, 217, 109, 449, 241, 361, 353, 1303, 239, 149, 73

Offset: 2

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.
For n>1, T_k has a primitive cycle of length n which includes 1 when k = A036563(n) = 2^n-3. So a(n) <= 2^n-3.

Geoffrey H. Morley, Table of n, a(n) for n = 2..3908

Cf. A226609, A226661, A226676.

A226615 Length of the Collatz-like 3x+k cycle associated with A226614(n).

Original entry on oeis.org

2, 3, 6, 4, 7, 5, 20, 11, 12, 28, 6, 38, 44, 48, 72, 18, 106, 29, 75, 7, 37, 14, 10, 42, 72, 66, 10, 68, 38, 58, 72, 8, 43, 110, 22, 33, 68, 29, 42, 71, 13, 46, 121, 28, 182, 200, 47, 9, 21, 60, 108, 28, 156, 19, 22, 85, 79, 151, 62, 56, 71, 60, 78, 226, 104, 192

Offset: 1

Geoffrey H. Morley, Aug 02 2013

nonn

Geoffrey H. Morley, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

PARI
```
See A226614.
```

a(n) = A226609(A226612(floor(A226614(n)/3)+1)).

For n>227, this formula requires terms beyond A226609(10000).

A226616 Smallest positive integer k for which 1 is in a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 13, 29, 11, 17, 253, 509, 145, 43, 55, 355, 137, 1129, 1007, 131069, 97, 643, 41, 553, 281, 8388605, 4069, 4793489, 3817, 1843, 59, 113, 1301, 2155, 9397, 289, 131153, 3247, 949, 127, 77

Offset: 2

Geoffrey H. Morley, Jul 02 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.
For n>1, T_k has a primitive cycle of length n which includes 1 when k = A036563(n) = 2^n-3. So a(n) <= 2^n-3.

Cf. A226607, A226609, A226617, A226618.

Showing 1-10 of 10 results.

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.

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A226610 Irregular array read by rows. a(n) is the number of odd elements in the primitive 3x+k cycle associated with A226607(n).

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A226625 Irregular array read by rows. a(n) is the length of 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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A226618 Irregular array read by rows in which row n lists the positive integers k in ascending order for which 1 is in a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

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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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A226660 Smallest positive integer k with a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

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A226615 Length of the Collatz-like 3x+k cycle associated with A226614(n).

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A226616 Smallest positive integer k for which 1 is in a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

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