A226629 -id:A226629 - cp's OEIS frontend

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.

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.

A226623 Irregular array read by rows in which row n lists the smallest elements, in ascending order, of conjecturally all primitive cycles of positive integers under iteration by the Collatz-like 3x-k function, where k = A226630(n).

Original entry on oeis.org

1, 5, 17, 19, 65, 73, 115, 2263, 2359, 2743, 2963, 3091, 3415, 3743, 4819, 113, 109, 95, 65, 989, 1153, 1165, 293, 511, 505, 625, 769, 211, 227, 251, 311, 1085, 2089, 7471, 10883, 13963, 15875, 16099, 1291, 1355, 1367, 1495, 1931, 2059, 2123, 2203, 2219, 2251

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, where k 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.

			The irregular array starts:
(k=1)  1, 5, 17;
(k=11) 19;
(k=17) 65, 73;
(k=19) 115;
a(4)=19 is the smallest number in the 3x-11 cycle {19,23,29,38}.

Geoffrey H. Morley, Rows 1..280 of array, flattened
J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arith. 56 (1990), 33-53.

Row n begins with a(A226628(n)) and has length A226629(n).
The smallest starting value whose trajectory includes a(n) is A226627(n). The cycle associated with a(n) has length A226625(n) and A226626(n) odd elements of which A226624(n) is the largest
Cf. A226607, A226681, A226682.

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.

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.

A226624 Irregular array read by rows. a(n) is the largest element in the primitive Collatz-like 3x-k cycle associated with A226623(n).

Original entry on oeis.org

1, 7, 91, 29, 179, 143, 505, 17033, 16793, 15497, 31613, 19589, 25781, 15845, 12137, 2011, 311, 517, 103, 19031, 24623, 8339, 811, 2609, 7387, 2995, 18275, 601, 493, 421, 1577, 74611, 13699, 1793597, 275693, 177521, 226769, 144881

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1, 7, 91;
(k=11) 29;
(k=17) 179, 143;
(k=19) 505;
a(4)=29 is the largest element 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 A226626(n) odd elements.
Cf. A226608, A226627, A226683, A226684, A226685.

A226627 Irregular array read by rows. a(n) is the smallest starting value of a T_k trajectory that includes A226623(n), where T_k is the Collatz-like 3x-k function associated with A226623(n).

Original entry on oeis.org

1, 5, 17, 19, 33, 73, 51, 2263, 2359, 2451, 1671, 2463, 1719, 2367, 4819, 89, 85, 63, 65, 685, 397, 1165, 293, 507, 369, 449, 769, 147, 227, 251, 247, 1085, 777, 7471, 7299, 11811, 5379, 8115, 267, 1355, 1367, 1043, 587, 779, 2123, 827, 2219, 843, 1611, 1707

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1, 5, 17;
(k=11) 19;
(k=17) 33, 73;
(k=19) 51;
a(5)=33 is the smallest starting value for a 3x-17 trajectory that includes A226623(5)=65. The trajectory is {33,41,53,71,98,49,65,...}.

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 A226626(n) odd elements of which A226624(n) is the largest.
Cf. A226611.

A226679 Record-breaking values, for increasing k = A226630(n), of the conjectured number of primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

Original entry on oeis.org

3, 8, 35, 136, 171, 2908, 6326

Offset: 1

Geoffrey H. Morley, Jun 15 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.

k = A226680(n).

Cf. A226629, A226663.

Definition clarified by Geoffrey H. Morley, Jun 23 2013

A226678 Smallest positive integer (or 0 if no such k) with conjecturally exactly n primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

Original entry on oeis.org

11, 17, 1, 385, 131, 193, 641, 23, 217, 7775, 57095, 3689, 1163, 14185, 8533, 467, 46199, 20143, 87089, 15217, 973, 134809, 14279

Offset: 1

Geoffrey H. Morley, Jun 25 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 and k is in A226630.
Conjecture: a(n)>0 for all n.

Cf. A226629, A226662.

cp's OEIS Frontend

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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Formula

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

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A226623 Irregular array read by rows in which row n lists the smallest elements, in ascending order, of conjecturally all primitive cycles of positive integers under iteration by the Collatz-like 3x-k function, where k = A226630(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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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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A226624 Irregular array read by rows. a(n) is the largest element in the primitive Collatz-like 3x-k cycle associated with A226623(n).

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A226627 Irregular array read by rows. a(n) is the smallest starting value of a T_k trajectory that includes A226623(n), where T_k is the Collatz-like 3x-k function associated with A226623(n).

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A226679 Record-breaking values, for increasing k = A226630(n), of the conjectured number of primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

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Extensions

A226678 Smallest positive integer (or 0 if no such k) with conjecturally exactly n primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

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