A226612 -id:A226612 - 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.

A226613 a(n) is the conjectured number of primitive cycles of positive integers under iteration by the Collatz-like 3x+k function, where n=floor(k/3)+1.

Original entry on oeis.org

1, 5, 1, 2, 9, 2, 1, 3, 2, 4, 1, 2, 3, 1, 1, 7, 1, 1, 3, 7, 2, 1, 1, 7, 3, 1, 4, 3, 1, 1, 3, 3, 2, 7, 2, 1, 1, 1, 2, 5, 2, 4, 2, 3, 2, 5, 1, 3, 3, 2, 2, 1, 1, 4, 2, 3, 2, 2, 7, 1, 3, 1, 2, 3, 4, 1, 2, 2, 1, 4, 1, 3, 2, 1, 2, 1, 8, 19, 3, 4, 2, 2, 6, 2, 3, 3, 7, 3

Offset: 1

Geoffrey H. Morley, Jun 13 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.

Geoffrey H. Morley, Table of n, a(n) for n = 1..6667
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). [Table 1 on page 19 gives a(1) to a(500).]

a(n) is the number of terms in the n-th row of A226607 to A226611.

Cf. A226629, A226663.

a(n) = A226612(n+1) - A226612(n).

A226628 Index of the first element of row n of A226623.

Original entry on oeis.org

1, 4, 5, 7, 8, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 31, 32, 34, 39, 74, 76, 77, 79, 80, 86, 95, 231, 232, 233, 237, 239, 240, 241, 257, 260, 268, 276, 277, 286, 287, 289, 290, 306, 313, 322, 323, 324, 325, 351, 372, 385, 388, 392, 395, 397, 399, 437

Offset: 1

Geoffrey H. Morley, Jun 13 2013

nonn

Geoffrey H. Morley, Table of n, a(n) for n = 1..600

a(n) is also the index of the first element of row n in A226624 to A226627.

Cf. A226612, A226629, A226631, A226632.

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.

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.

A226614 Positive integers k for which 1 is in a cycle of integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 11, 13, 17, 29, 41, 43, 55, 59, 61, 77, 79, 91, 95, 97, 107, 113, 119, 125, 127, 137, 145, 155, 185, 193, 203, 209, 215, 239, 247, 253, 257, 275, 281, 289, 317, 329, 335, 353, 355, 407, 437, 445, 473, 493, 499, 509, 553, 559, 593, 629, 637, 643, 673, 697

Offset: 1

Geoffrey H. Morley, Aug 02 2013

nonn

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. GCD(k,6)=1.
When k=2^m-3, T_k has a cycle containing 1. Hence the sequence is infinite.
a(n) is in the sequence if and only if A226607(A226612(floor(a(n)/3)+1)) = 1.
Trivially, members of the sequence are not divisible by 2 or 3. Of the first 10^4 members, only 1,066 are squareful, which is about one third of the expected density. - Ralf Stephan, Aug 05 2013

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

Cf. A226615-A226618.

\\ 5.5 hours (2.33 Ghz Intel Core 2)
{k=1; n=1;
until(n>10000, x=1; y=1; len=0;
  until(x==y, if(x%2==0, x=x/2, x=(3*x+k)/2);
    if(y%2==0, y=y/2, y=(3*y+k)/2);
    if(y%2==0, y=y/2, y=(3*y+k)/2); len++);
  if(x==1, write("b226614.txt",n," ",k);
    write("b226615.txt",n," ",len); n++);
  k+=(k+3)%6)}

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).

Showing 1-9 of 9 results.

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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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A226613 a(n) is the conjectured number of primitive cycles of positive integers under iteration by the Collatz-like 3x+k function, where n=floor(k/3)+1.

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A226628 Index of the first element of row n of A226623.

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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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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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A226614 Positive integers k for which 1 is in a cycle of integers under iteration by the Collatz-like 3x+k function.

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PARI

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

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