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A262670 Consider the Farey sequence of order n, F_n, and that the average distance between any two adjacent pairs in F_n is 1/A002088(n). Then a(n) is the number of adjacent pairs whose difference is the average.

%I A262670 #13 Feb 16 2025 08:33:27
%S A262670 0,1,2,0,2,2,2,0,0,2,0,2,0,0,0,2,2,0,2,2,0,2,0,0,2,2,0,2,0,2,0,2,0,0,
%T A262670 0,0,2,2,2,0,0,0,0,0,0,0,2,2,0,2,2,2,0,2,2,2,0,0,0,0,4,0,0,0,6,0,0,2,
%U A262670 4,2,0,2,0,0,2,2,0,0,2,0,0,0,0,2,0,0,2,0,2,0,2,2,2,2,0,2,2,0,0,0,0,0,0,0,0
%N A262670 Consider the Farey sequence of order n, F_n, and that the average distance between any two adjacent pairs in F_n is 1/A002088(n). Then a(n) is the number of adjacent pairs whose difference is the average.
%C A262670 Because the Farey fractions are symmetrical about 1/2 for n > 1, a(n) is always even.
%C A262670 First occurrence of k by index, or -1 if no such occurrence exists: 0, 1, 2, -1, 60, -1, 64, -1, 207, -1, 1047, -1, 1084, -1, ..., .
%C A262670 Where 0 occurs: 0, 3, 7, 8, 10, 12, 13, 14, 17, 20, 22, 23, 26, 28, 30, 32, 33, ..., ;
%C A262670 Where 2 occurs: 2, 4, 5, 6, 9, 11, 15, 16, 18, 19, 21, 24, 25, 27, 29, 31, 36, 37, 38, ..., ;
%C A262670 Where 4 occurs: 60, 68, 120, 129, 148, 158, 159, 168, 180, 216, 225, 231, 239, 241, 249, ..., ;
%C A262670 Where 6 occurs: 65, 227, 401, 403, 492, 600, 616, 780, 861, 862, 865, 967, 1019, 1054, ..., ;
%C A262670 Where 8 occurs: 208, 1210, 1367, 1803, 1804, 1841, 1866, 2397, 2864, 3281, 3443, 3724, ..., ;
%C A262670 Where 10 occurs: 1048, 1094, 1632, 1949, 2269, 2571, 2710, 3365, 3555, 3558, 3613, 3939, ..., ;
%C A262670 Where 12 occurs: 1085, 1358, 2541, 3251, 4411, ..., ;
%C A262670 Where 18 occurs: 4830, ..., ;
%C A262670 For the first 5001 terms: 3315 zeros, 1 one, 1138 twos, 414 fours, 96 sixes, 19 eights, 12 tens, 5 twelves and 1 eighteen.
%D A262670 Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Chapter XVI, "Farey Tails", Dover Books, NY, 1966, pgs 168-172.
%H A262670 Cut the Knot, <a href="http://www.cut-the-knot.org/blue/Farey.shtml">Farey Series</a>.
%H A262670 The University of Surrey, Math Dept., <a href="https://r-knott.surrey.ac.uk/Fractions/fareySB.html">Fractions in the Farey Series and the Stern-Brocot Tree</a>.
%H A262670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FareySequence.html">Farey Sequence</a>.
%H A262670 Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey Sequence</a>.
%e A262670 a(5) = 2. F_5 = {0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1} and the first forward difference is {1/5, 1/20, 1/12, 1/15, 1/10, 1/10, 1/15, 1/12, 1/20, 1/5}. The average distance is 1/10 since A002088(5) = 10 which is also the number of adjacent pairs, a/b & c/d.
%t A262670 f[n_] := Block[{diff = Differences@ Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}], ave = 1/Sum[ EulerPhi[ m], {m, n}]}, {Length@ Select[diff, ave < # &], Length@ Select[diff, ave == # &], Length@ Select[diff, ave > # &]}]; Array[f, 65]
%Y A262670 Cf. A002088, A005728, A006843, A262669.
%K A262670 nonn
%O A262670 0,3
%A A262670 _Robert G. Wilson v_, Nov 09 2015