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A001000
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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.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 (davidwwilson(AT)comcast.net), Aug 30 2007
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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
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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 (davidwwilson(AT)comcast.net), Aug 30 2007
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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.
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CROSSREFS
| Sequence in context: A024785 A069866 A125772 * A094947 A092621 A188809
Adjacent sequences: A000997 A000998 A000999 * A001001 A001002 A001003
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KEYWORD
| nonn,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Christopher Carl Heckman pointed out that the old definition was incomplete and Clark Kimberling supplied a revised definition Feb 18 2004.
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