OFFSET
1,1
COMMENTS
It suffices for (a/b, c/d) to range through the consecutive pairs of Farey fractions of order n.
This is the same sequence (apart from the initial term) as A071111. The identity of these two sequences was first proved by Rustem Aidagulov and a detailed version of the proof can be found in the Alekseyev link below.
For sets of real numbers S and T, let S be a divider of T if some element of S lies strictly between any two distinct elements of T. Let Fence(n) = {a/n : a in Z}, Recip(n) = {1/b : 1 <= b <= n} Farey(n) = {a/b : a in Z, 1 <= b <= n}. Then a(n) is the smallest k such that Fence(k) is a divider of Recip(n) and also the smallest k such that Fence(k) is a divider of Farey(n), as shown by S. Rustem Aidagulov. - David W. Wilson, Aug 30 2007
Suppose that S is a set of 2 or more real numbers. The least m such that for every r and s in S there is an integer k such that r < k/m < s is the first separator of S. The least m such that for every r and s in S there exists an integer k such that r < k/m < (k+1)/m < s is the second separator of S, and so on.
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For example, A001000 gives first separators for the sets S(n) = {0,1/2,1/3,...,1/n}. In the following guide, the set S(n) consists of numbers given by the shown formula for h = 1, 2, ..., n; F = A000045 (Fibonacci numbers), and r = (1+sqrt(5))/2 (golden ratio).
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S(n) ................... 1st separators
Farey fractions ........ A001000
1/h .................... A071111
1/(2*h-1) .............. A024819
1/(2*h) ................ A024820
1/sqrt(h) .............. A024821
1/(3*h-2) .............. A024822
1/(3*h-1) .............. A024823
1/(3*h) ................ A024824
1/(4*h) ................ A024825
1/C(n+1,2) ............. A024826
1/h^2 .................. A024827
h/(1+h^2) .............. A024828
F(2*h-1)/F(2*h)......... A024829
F(2*h)/F(2*h+1) ........ A024830
F(2*h+1)/F(2*h+2)....... A024831
pi/2 - arctan(h) ....... A024832
|F(h+1)-r*F(h)| ........ A024849
fr. parts, h*sqrt(2) ... A214921
fr. parts, h*r ......... A214964
fr. parts, h*e ......... A214965
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S(n) ............. 2nd separators
1/h .............. A024833
1/(2*h-1) ........ A024834
1/(2*h) .......... A024835
1/(3*h-2) ........ A024836
1/(3*h-1) ........ A024837
1/(3*h) .......... A024838
1/(4*h) .......... A024839
...
S(n) ............. 3rd separators
1/h .............. A024840
1/(2*h-1) ........ A024841
1/(2*h) .......... A024842
...
S(n) ............. 4th separators
1/h .............. A024843
1/(2*h-1) ........ A024844
1/(2*h) .......... A024845
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S(n) ............. 5th separators
1/h .............. A024846
1/(2*h-1) ........ A024847
- Clark Kimberling, Aug 07 2012
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
For n >= 2, a(n) = (n-[r])(n-[r+1/2])+1, where r = sqrt(4n-7), [x] = greatest integer <= x. - David W. Wilson, Aug 30 2007
EXAMPLE
The Farey fractions of order 4, 0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1, are separated by the fractions k/7: 0/1 < 1/7 <1/4 < 2/7 < 1/3 < 3/7 < 1/2 < 4/7 < 2/3 <5/7 < 3/4 <6/7 < 1 and 7 is the least m for which at least one k/m lies strictly between each pair of Farey fractions.
MATHEMATICA
(* The following program generates a northwest corner of an array in which row k shows the least k-th separator of the set {1/h : h = 1, 2, ..., n}. *)
leastSeparatorS[seq_, s_] := Module[{n = 1},
Table[While[Or @@ (n #[[1]] <=
s + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
TableForm[Map[leastSeparatorS[1/Range[15], #] &, Range[10]]]
(* Peter J. C. Moses, Aug 07 2012 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Incompleteness of old definition pointed out by Christopher Carl Heckman, and revised definition supplied by Clark Kimberling, Feb 18 2004
Definition of separator, guide to related sequences, and Mathematica program added by Clark Kimberling, Aug 07 2012
STATUS
approved