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A355616 a(n) is the number of distinct lengths between consecutive points of the Farey sequence of order n.

%I A355616 #31 Jul 16 2022 12:04:18
%S A355616 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,
%T A355616 100,110,115,132,133,150,158,165,169,182,187,208,213,222,228,252,254,
%U A355616 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
%N A355616 a(n) is the number of distinct lengths between consecutive points of the Farey sequence of order n.
%C A355616 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.
%e A355616 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.
%t A355616 a[n_] := FareySequence[n] // Differences // Union // Length;
%t A355616 Table[a[n], {n, 1, 70}] (* _Jean-François Alcover_, Jul 16 2022 *)
%o A355616 (Python)
%o A355616 from fractions import Fraction
%o A355616 from itertools import chain
%o A355616 def compute(n):
%o A355616     marks = [[(a, b) for a in range(0, b + 1)] for b in range(1, n + 1)]
%o A355616     marks = sorted(set([Fraction(a, b) for a, b in chain(*marks)]))
%o A355616     dist = [(y - x) for x, y in zip(marks, marks[1:])]
%o A355616     return len(set(dist))
%o A355616 (PARI) vp(n) = my(list = List()); for (k=1, n, for (i=0, k, listput(list, i/k))); vecsort(list,,8);
%o A355616 a(n) = my(v=vp(n)); #Set(vector(#v-1, k, abs(v[k+1]-v[k]))); \\ _Michel Marcus_, Jul 10 2022
%Y A355616 Cf. A006842/A006843 (Farey sequences).
%Y A355616 Cf. A005728 (number of distinct points).
%K A355616 nonn
%O A355616 1,3
%A A355616 _Travis Hoppe_, Jul 09 2022