Search: id:A001000
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%I A001000
%S A001000 2,3,5,7,13,17,26,31,43,57,65,82,101,111,133,157,183,197,226,257,290,307,
%T A001000 343,381,421,463,485,530,577,626,677,703,757,813,871,931,993,1025,1090,
%U A001000 1157,1226,1297,1370,1407,1483,1561,1641,1723,1807,1893,1937,2026,2117
%N 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.
%C A001000 It suffices for (a/b, c/d) to range through the consecutive pairs of 
               Farey fractions of order n.
%C A001000 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.
%C A001000 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
%H A001000 T. D. Noe, <a href="b001000.txt">Table of n, a(n) for n=1..1000</a>
%H A001000 Max Alekseyev, <a href="a071111.txt">Proof that A001000 and A071111 are 
               essentially the same sequence</a>
%F A001000 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
%e A001000 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.
%Y A001000 Sequence in context: A024785 A069866 A125772 this_sequence A094947 A092621 
               A152449
%Y A001000 Adjacent sequences: A000997 A000998 A000999 this_sequence A001001 A001002 
               A001003
%K A001000 nonn,nice
%O A001000 1,1
%A A001000 Clark Kimberling (ck6(AT)evansville.edu)
%E A001000 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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