



2, 3, 3, 5, 4, 5, 4, 8, 7, 7, 5, 8, 7, 7, 5, 13, 11, 12, 9, 11, 10, 9, 6, 13, 11, 12, 9, 11, 10, 9, 6, 21, 18, 19, 14, 19, 17, 16, 11, 18, 15, 17, 13, 14, 13, 11, 7, 21, 18, 19, 14, 19, 17, 16, 11, 18, 15, 17, 13, 14, 13, 11, 7, 34, 29, 31, 23, 30, 27, 25, 17, 31, 26, 29, 22, 25, 23, 20, 13, 29, 25, 26, 19, 27, 24, 23, 16, 23
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OFFSET

1,1


COMMENTS

The terms (n>0) may be written as a leftjustified array with rows of length 2^m, m >= 0:
2,
3, 3,
5, 4, 5, 4,
8, 7, 7, 5, 8, 7, 7, 5,
13,11,12, 9,11,10, 9, 6,13,11,12, 9,11,10, 9, 6,
21,18,19,14,19,17,16,11,18,15,17,13,14,13,11, 7,21,18,19,14,19,17,...
All columns have the Fibonacci sequence property: a(2^(m+2) + k) = a(2^(m+1) + k) + a(2^m + k), m ≥ 0 , 0 ≤ k < 2^m (empirical observations).
The terms (n>0) may also be written as a leftjustified array with rows of length 2^m, m ≥ 0:
2,
3, 3,
5, 4, 5, 4,
8, 7, 7, 5, 8, 7, 7, 5,
13,11,12, 9,11,10, 9, 6,13,11,12, 9,11,10, 9, 6,
..., 18,15,17,13,14,13,11, 7,21,18,19,14,19,17,16,11,18,15,17,13,14,13,11, 7,
Each column is an arithmetic sequence. The differences of the arithmetic sequences give the sequence A071585: a(2^(m+1)1k)  a(2^m1k) = A071585(k), m ≥ 0 , 0 ≤ k < 2^m .
n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (Franklin T. AdamsWatters' comment), that is the sequence obtained by adding numerator and denominator in the CalkinWilf enumeration system of positive rationals. A245325(n)/A245326(n) is also an enumeration system of all positive rationals, and in each level m >= 0 (ranks between 2^m and 2^(m+1)1) rationals are the same in both systems. Thus a(n) has the same terms in each level as A007306.
The same property occurs in all numerator+denominator sequences of enumeration systems of positive rationals, as, for example, A007306 (A007305+A047679), A071585 (A229742+A071766), A086592 (A020650+A020651), A268087 (A162909+A162910).
a(n) = A273493(A059893(n)), a(A059893(n)) = A273493(n), n > 0.  Yosu Yurramendi, May 30 2017


LINKS

Table of n, a(n) for n=1..88.


CROSSREFS

Cf. A007306, A268087, A071585, A086592, A273493
Sequence in context: A210874 A244796 A080391 * A116922 A086898 A123031
Adjacent sequences: A273491 A273492 A273493 * A273495 A273496 A273497


KEYWORD

nonn


AUTHOR

Yosu Yurramendi, May 23 2016


STATUS

approved



