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A355616
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a(n) is the number of distinct lengths between consecutive points of the Farey sequence of order n.
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0
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1, 1, 2, 3, 5, 6, 9, 11, 14, 15, 21, 23, 29, 31, 34, 38, 48, 49, 59, 63, 67, 71, 83, 86, 97, 100, 110, 115, 132, 133, 150, 158, 165, 169, 182, 187, 208, 213, 222, 228, 252, 254, 280, 287, 297, 304, 331, 337, 362, 367, 379, 387, 418, 423, 437, 450, 464, 472, 509, 513, 548, 556, 573, 589, 608, 611, 652, 665, 681, 685
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OFFSET
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1,3
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COMMENTS
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The Farey sequence of order n (row n of A006842/A006843) is the set of points x/y on the unit line where 1 <= y <= n and 0 <= x <= y.
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LINKS
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Table of n, a(n) for n=1..70.
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EXAMPLE
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For n=5, the Farey sequence (completely reduced fractions) is [0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1]. The distinct lengths between consecutive points are {1/5, 1/20, 1/12, 1/15, 1/10} so a(5) = 5.
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MATHEMATICA
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a[n_] := FareySequence[n] // Differences // Union // Length;
Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Jul 16 2022 *)
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PROG
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(Python)
from fractions import Fraction
from itertools import chain
def compute(n):
marks = [[(a, b) for a in range(0, b + 1)] for b in range(1, n + 1)]
marks = sorted(set([Fraction(a, b) for a, b in chain(*marks)]))
dist = [(y - x) for x, y in zip(marks, marks[1:])]
return len(set(dist))
(PARI) vp(n) = my(list = List()); for (k=1, n, for (i=0, k, listput(list, i/k))); vecsort(list, , 8);
a(n) = my(v=vp(n)); #Set(vector(#v-1, k, abs(v[k+1]-v[k]))); \\ Michel Marcus, Jul 10 2022
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CROSSREFS
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Cf. A006842/A006843 (Farey sequences).
Cf. A005728 (number of distinct points).
Sequence in context: A113238 A104214 A349523 * A046657 A102825 A070991
Adjacent sequences: A355613 A355614 A355615 * A355617 A355618 A355619
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KEYWORD
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nonn
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AUTHOR
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Travis Hoppe, Jul 09 2022
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STATUS
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approved
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