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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.
3
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, 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, 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
OFFSET
0,3
COMMENTS
Because the Farey fractions are symmetrical about 1/2 for n > 1, a(n) is always even.
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, ..., .
Where 0 occurs: 0, 3, 7, 8, 10, 12, 13, 14, 17, 20, 22, 23, 26, 28, 30, 32, 33, ..., ;
Where 2 occurs: 2, 4, 5, 6, 9, 11, 15, 16, 18, 19, 21, 24, 25, 27, 29, 31, 36, 37, 38, ..., ;
Where 4 occurs: 60, 68, 120, 129, 148, 158, 159, 168, 180, 216, 225, 231, 239, 241, 249, ..., ;
Where 6 occurs: 65, 227, 401, 403, 492, 600, 616, 780, 861, 862, 865, 967, 1019, 1054, ..., ;
Where 8 occurs: 208, 1210, 1367, 1803, 1804, 1841, 1866, 2397, 2864, 3281, 3443, 3724, ..., ;
Where 10 occurs: 1048, 1094, 1632, 1949, 2269, 2571, 2710, 3365, 3555, 3558, 3613, 3939, ..., ;
Where 12 occurs: 1085, 1358, 2541, 3251, 4411, ..., ;
Where 18 occurs: 4830, ..., ;
For the first 5001 terms: 3315 zeros, 1 one, 1138 twos, 414 fours, 96 sixes, 19 eights, 12 tens, 5 twelves and 1 eighteen.
REFERENCES
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.
LINKS
Cut the Knot, Farey Series.
The University of Surrey, Math Dept., Fractions in the Farey Series and the Stern-Brocot Tree.
Eric Weisstein's World of Mathematics, Farey Sequence.
Wikipedia, Farey Sequence.
EXAMPLE
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.
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Nov 09 2015
STATUS
approved