cp's OEIS Frontend

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.)

%I A226605 #8 Apr 06 2020 22:13:54
%S A226605 -1,0,1,-5,1,-19,5,1,-65,19,23,5,7,1,-211,-65,-73,19,23,31,1,7,1,-665,
%T A226605 -211,-227,65,-251,73,89,19,85,101,23,31,47,5,37,1,11,1,-2059,-665,
%U A226605 -697,211,-745,227,259,13,251,283,73,331,89,121,19,319,17,101,19,23
%N 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.)
%C A226605 A cycle is irreducible if it is not a concatenation of copies of a shorter cycle.
%C A226605 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.
%C A226605 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.
%C A226605 The element with numerator a(n) has denominator A226606(n), as does every element in the same cycle.
%C A226605 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.
%H A226605 Geoffrey H. Morley, <a href="/A226605/b226605.txt">Rows 1..16 of array, flattened</a>
%H A226605 J. C. Lagarias, <a href="http://pldml.icm.edu.pl:80/mathbwn/element/bwmeta1.element.bwnjournal-article-aav56i1p33bwm?q=bwmeta1.element.bwnjournal-number-aa-1990-56-1&amp;qt=CHILDREN-STATELESS">The set of rational cycles for the 3x+1 problem,</a> Acta Arith. 56 (1990), 33-53. - See Table 2.2 on page 39.
%F A226605 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).
%e A226605 -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.
%Y A226605 There are A001037(n) terms in row n.
%K A226605 sign,frac,tabf
%O A226605 1,4
%A A226605 _Geoffrey H. Morley_, Jun 27 2013