A001000 a(n) = least m such that if a/b < c/d where a,b,c,d are integers in [0,n], then a/b < k/m < c/d for some integer k.

2, 3, 5, 7, 13, 17, 26, 31, 43, 57, 65, 82, 101, 111, 133, 157, 183, 197, 226, 257, 290, 307, 343, 381, 421, 463, 485, 530, 577, 626, 677, 703, 757, 813, 871, 931, 993, 1025, 1090, 1157, 1226, 1297, 1370, 1407, 1483, 1561, 1641, 1723, 1807, 1893, 1937, 2026, 2117

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• nice

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.

...

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).

...

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

...

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

...

S(n) ............. 5th separators

1/h .............. A024846

1/(2*h-1) ........ A024847

- Clark Kimberling, Aug 07 2012

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Max Alekseyev, Proof that A001000 and A071111 are essentially the same sequence

Crossrefs

Cf. A071111.

Programs

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 *)

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

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