A226606 -id:A226606 - cp's OEIS frontend

A226619 Irregular array read by rows in which row n lists the integers k, in ascending order, for which there is a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

-1, 1, 1, -1, 5, -11, 7, 13, -49, 5, 23, 29, -179, -17, 11, 37, 55, 61, -601, -115, 17, 47, 101, 119, 125, -1931, -473, 13, 25, 35, 175, 229, 247, 253, -6049, -1675, -217, -31, 97, 269, 431, 485, 503, 509, -18659, -5537, -1163, -791, 59, 71, 145, 203, 295, 781, 943, 997, 1015, 1021

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.
We associate the cycle {0} with k = A226606(2) = 1.
For n>1 the first term of row n is 2^n-3^(n-1), and the last term is A036563(n) = 2^n-3.

			The irregular array starts:
-1, 1;
1;
-1, 5;
-11, 7, 13;
-49, 5, 23, 29; ...

Geoffrey H. Morley, Rows 1..26 of array, flattened

Cf. A226605, A226606, A226618, A226620, A226621.

A226605 Irregular array read by rows of numerators in which row n has one numerator from each irreducible cycle of n rational numbers under iteration by the 3x+1 function. (See Comments for selection and order of numerators.)

Original entry on oeis.org

-1, 0, 1, -5, 1, -19, 5, 1, -65, 19, 23, 5, 7, 1, -211, -65, -73, 19, 23, 31, 1, 7, 1, -665, -211, -227, 65, -251, 73, 89, 19, 85, 101, 23, 31, 47, 5, 37, 1, 11, 1, -2059, -665, -697, 211, -745, 227, 259, 13, 251, 283, 73, 331, 89, 121, 19, 319, 17, 101, 19, 23

Offset: 1

Geoffrey H. Morley, Jun 27 2013

A cycle is irreducible if it is not a concatenation of copies of a shorter cycle.
The 3x+1 function T, on rational numbers in their lowest terms with a positive odd denominator, is defined by T(x) = x/2 if x's numerator is even, T(x) = (3x+1)/2 if x's numerator is odd.
Each numerator in a row is the first in the cyclic permutation with the lexicographically largest parity vector of numerators mod 2. The row lists these numerators in descending lexicographic order of the parity vectors.
The element with numerator a(n) has denominator A226606(n), as does every element in the same cycle.
a(n) is often the numerator with the least absolute value of the numerators in the cycle. a(20) and a(36) are the only exceptions in the first 7 rows.

			-1, 0, 1, -5, 1/5, -19/11, 5/7, 1/13, ... = A226605/A226606 for parity vectors 1, 0, 10, 110, 100, 1110, 1100, 1000, ... For example, the numerators of the rational cycle {-19/11,-23/11,-29/11,-38/11} have parity vector 1110.

Geoffrey H. Morley, Rows 1..16 of array, flattened
J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arith. 56 (1990), 33-53. - See Table 2.2 on page 39.

There are A001037(n) terms in row n.

If v(0) to v(m-1) are the bits of A102659(n), when 2's are replaced by 0's, then a(n) = N(n)/GCD(N(n),D(n)) where D(n) = 2^m - 3^(v(0)+...+v(m-1)) and N(n) = Sum_{j=0 to m-1} (2^j)(3^(v(j+1)+...+v(m-1)))v(j).

cp's OEIS Frontend

A226619 Irregular array read by rows in which row n lists the integers k, in ascending order, for which there is a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

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