

A093875


Denominators in Kepler's tree of harmonic fractions.


8



1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 8, 8, 5, 5, 7, 7, 7, 7, 8, 8, 6, 6, 9, 9, 10, 10, 11, 11, 9, 9, 12, 12, 11, 11, 13, 13, 6, 6, 9, 9, 10, 10, 11, 11, 9, 9, 12, 12, 11, 11, 13, 13, 7, 7, 11, 11, 13, 13, 14, 14, 13, 13, 17, 17, 15, 15, 18, 18, 11, 11, 16, 16
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OFFSET

1,2


COMMENTS

Form a tree of fractions by beginning with 1/1 and then giving every node i/j two descendants labeled i/(i+j) and j/(i+j).
It appears that A071585 is a bisection of this sequence, which itself is a bisection of A093873.  Yosu Yurramendi, Jan 09 2016


LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for fraction trees


FORMULA

a(n) = a([n/2]) + A093873([n/2]).
Conjecture of the comment in detail: a(2n+1) = a(2n), n > 0; a(2n+1) = A071585(n), n >= 0; a(2n) = A071585(n), n > 0.  Yosu Yurramendi, Jun 22 2016
a(2n)  A093873(2n) = a(n), n > 0; a(2n+1)  A093873(2n+1) = A093873(n), n > 0.  Yosu Yurramendi, Jul 23 2016
From Yosu Yurramendi, Jul 25 2016: (Start)
a(2^m) = m+1, m >= 0; a(2^m + 2) = 2m  1, m >= 1; a(2^m  1) = A000045(m+2), m >= 1.
a(2^(m+1) + k)  a(2^m + k) = a(k), m > 0, 0 <= k < 2^m. For k=0, a(0) = 1 is needed.
a(2^(m+2)  k  1) = a(2^(m+1)  k  1) + a(2^m  k  1), m >= 0, 0 <= k < 2^m. (End)


EXAMPLE

The first few fractions are:
1 1 1 1 2 1 2 1 3 2 3 1 3 2 3 1 4 3 4 2 5 3 5 1 4 3 4 2 5 3 5
                               ...
1 2 2 3 3 3 3 4 4 5 5 4 4 5 5 5 5 7 7 7 7 8 8 5 5 7 7 7 7 8 8


MATHEMATICA

num[1] = num[2] = 1; den[1] = 1; den[2] = 2; num[n_?EvenQ] := num[n] = num[n/2]; den[n_?EvenQ] := den[n] = num[n/2] + den[n/2]; num[n_?OddQ] := num[n] = den[(n1)/2]; den[n_?OddQ] := den[n] = num[(n1)/2] + den[(n1)/2]; A093875 = Table[den[n], {n, 1, 83}] (* JeanFrançois Alcover, Dec 16 2011 *)


CROSSREFS

The numerators are in A093873. Usually one only considers the lefthand half of the tree, which gives the fractions A020651/A086592. See A086592 for more information, references to Kepler, etc.
Cf. A071585, A093873
Sequence in context: A101402 A156251 A116458 * A266193 A114214 A321318
Adjacent sequences: A093872 A093873 A093874 * A093876 A093877 A093878


KEYWORD

nonn,easy,frac


AUTHOR

N. J. A. Sloane and Reinhard Zumkeller, May 24 2004


STATUS

approved



