A226686 -id:A226686 - cp's OEIS frontend

A226687 Integer k associated with the conjectured record-breaking length A226686(n) of primitive Collatz-like 3x-k cycles.

Original entry on oeis.org

1, 19, 23, 25, 113, 467, 697, 3377, 7153, 34595, 41807, 62675, 66781, 271591

Offset: 1

Geoffrey H. Morley, Jun 16 2013

nonn

Cf. A226671, A226688.

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.

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.

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

A229122 For odd m, let f(m) be the odd part of 3m+1. a(n) is the least positive number of f-iterations of 2n-1 to reach an odious number (A000069), or 0 if no such number of f-iterations exists.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 3, 2, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2

Offset: 1

Vladimir Shevelev, Oct 07 2013

nonn

Since 1 is odious number, the conjecture that all a(n) > 0 is a very weak form of the "3x+1" (Collatz) conjecture.

We conjecture that this sequence is unbounded.

			For n = 26, 2*n - 1 = 51; f(51) = 77 is evil; f(77) = 29 is evil; f(29) = 11 is odious, so a(26) = 3.

Cf, A226686, A226687.

Table[m = 2 n - 1; NestWhile[# + 1 &, 1, !OddQ[DigitCount[m = # / 2^IntegerExponent[#, 2] & [3 m + 1], 2][[1]]] &], {n, 100}] (* Peter J. C. Moses, Oct 13 2013 *)

More terms from Peter J. C. Moses

cp's OEIS Frontend

A226687 Integer k associated with the conjectured record-breaking length A226686(n) of primitive Collatz-like 3x-k cycles.

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A229122 For odd m, let f(m) be the odd part of 3m+1. a(n) is the least positive number of f-iterations of 2n-1 to reach an odious number (A000069), or 0 if no such number of f-iterations exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

cp's OEIS Frontend

A226687 Integer k associated with the conjectured record-breaking length A226686(n) of primitive Collatz-like 3x-k cycles.

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A229122 For odd m, let f(m) be the odd part of 3*m+1. a(n) is the least positive number of f-iterations of 2*n-1 to reach an odious number (A000069), or 0 if no such number of f-iterations exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A229122 For odd m, let f(m) be the odd part of 3m+1. a(n) is the least positive number of f-iterations of 2n-1 to reach an odious number (A000069), or 0 if no such number of f-iterations exists.