cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Geoffrey H. Morley, Jun 13 2013

Keywords

Examples

			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.

Links

Crossrefs

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.

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

Views

Author

Geoffrey H. Morley, Jun 13 2013

Keywords

Comments

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.

Examples

			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}.

Links

Crossrefs

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

Views

Author

Geoffrey H. Morley, Jun 13 2013

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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.

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

Views

Author

Geoffrey H. Morley, Jun 13 2013

Keywords

Examples

			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.

Links

Crossrefs

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

Views

Author

Geoffrey H. Morley, Jun 13 2013

Keywords

Examples

			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,...}.

Links

Crossrefs

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.

A226632 Fewest odd numbers in a primitive cycle of positive integers under iteration by the 3x-k function, where k=A226630(n).

Original entry on oeis.org

1, 3, 4, 11, 12, 22, 8, 6, 4, 14, 8, 11, 11, 44, 5, 12, 14, 24, 7, 8, 5, 43, 28, 19, 6, 12, 43, 10, 14, 28, 15, 20, 53, 6, 11, 11, 20, 8, 43, 12, 6, 14, 10, 14, 54, 8, 10, 14, 14, 7, 10, 12, 28, 9, 43, 14, 7, 25, 11, 10, 35, 7, 36, 32, 24, 28, 17, 24
Offset: 1

Views

Author

Geoffrey H. Morley, Jun 14 2013

Keywords

Comments

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.

Links

Formula

a(n) = min_{j=A226628(n) to A226628(n+1)-1} A226626(j).

A226677 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, 1, 11, 17, 115, 31, 1, 29, 1417, 371, 19, 23, 8977, 77, 431, 2465, 2069, 3299, 193, 451, 139, 25, 5233, 131, 1739, 10993, 3037, 121, 7061, 11329, 9479, 145, 2425, 46199, 1871, 217, 3551, 26183, 14083, 26281, 7237, 605, 181, 113, 3299, 11431, 119773, 2465
Offset: 1

Views

Author

Geoffrey H. Morley, Jul 05 2013

Keywords

Comments

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.

Links

Crossrefs

Cf. A226626, A226661, A226676.

A226689 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

7, 11, 12, 22, 44, 53, 54, 106, 108, 112, 113, 144, 159, 180
Offset: 1

Views

Author

Geoffrey H. Morley, Jun 19 2013

Keywords

Comments

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.
For n<15, but probably not for all n, k = A226687(n) and the smallest integer in the T_k cycle associated with a(n) is A226688(n).

Crossrefs

Cf. A226626, A226673.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.