A226616 -id:A226616 - cp's OEIS frontend

A226618 Irregular array read by rows in which row n lists the positive integers k in ascending order for which 1 is in a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

1, 5, 13, 29, 11, 61, 17, 125, 253, 509, 145, 203, 1021, 43, 2045, 55, 4093, 355, 1169, 8189, 137, 3275, 16381, 1129, 32765, 1007, 5957, 9361, 65533, 131069, 97, 52427, 262141, 643, 74897, 524285, 41, 1048573, 553, 28727, 110375, 2097149, 281, 673, 2075, 9731, 34663

Offset: 1

Geoffrey H. Morley, Jul 02 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.

			The irregular array starts:
1;
5;
13;
29;
11, 61;
17, 125; ...
Row 1 is empty.

Geoffrey H. Morley, Rows 2..26 of array, flattened

The first element in row n is A226616(n), and the last is A036563(n) = 2^n-3.

Cf. A226609, A226619.

A226617 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, n of which are odd including 1, under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 11, 43, 55, 643, 97, 673, 41, 1843, 329, 59, 113, 5603, 289, 6505, 77, 407, 127, 499, 79, 865, 749

Offset: 1

Geoffrey H. Morley, Jul 03 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.

			The cycle associated with a(1)=1 is {1,2}, with a(2)=11 is {1,7,16,8,4,2}, and with a(3)=43 is {1,23,56,28,14,7,32,16,8,4,2}.

Cf. A226607, A226610, A226616.

cp's OEIS Frontend

A226618 Irregular array read by rows in which row n lists the positive integers k in ascending order for which 1 is in a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

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