A226676 -id:A226676 - cp's OEIS frontend

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

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.

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

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.
Conjecture: a(n)>0 for all n.

Geoffrey H. Morley, Table of n, a(n) for n = 1..60

Cf. A226626, A226661, A226676.

cp's OEIS Frontend

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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