This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385974 #11 Jul 23 2025 23:48:13
%S A385974 1,7,7,23,23,47,47,87,87,107,107,179,179,179,179,275,275,347,347,447,
%T A385974 447,491,491,731,731,731,731,787,787,847,847,1071,1071,1071,1071,1359,
%U A385974 1359,1359,1359,1719,1719,1887,1887,2019,2019,2111,2111,2639,2639,2639,2639,2743,2743,2851,2851
%N A385974 a(n) = the number of reduced positive proper fractions p/q with phi(q) <= n.
%C A385974 This sequence groups the reduced positive proper fractions p/q in a manner analogous to the Farey sequence (A005728, which adds the endpoints 0/1 and 1/1). However, instead of limiting the size of the denominator (q <= n as is done in the Farey sequence), we limit phi(q) = A000010(q), where phi is the Euler totient function. The computation requires A014197(r) = the number of natural numbers q such that phi(q) = r.
%C A385974 When k > 0, a(2k+1) = a(2k), because there are no natural numbers whose Euler totient equals 2k+1.
%H A385974 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%F A385974 a(n) = -1 + Sum_{i=1..n} i*A014197(i).
%e A385974 a(4) enumerates the positive proper fractions whose denominator has Euler totient <= 4. The eligible denominators are 2; 3, 4, 6; 5, 8, 10, 12. The numerators are relatively prime to the denominators. Grouping the fractions by the order of the denominators above, we get: 1/2; 1/3, 2/3, 1/4, 3/4, 1/6, 5/6; 1/5, 2/5, 3/5, 4/5, 1/8, 3/8, 5/8, 7/8, 1/10, 3/10, 7/10, 9/10, 1/12, 5/12, 7/12, 11/12. Thus a(4) = 23.
%o A385974 (PARI) a(n) = sum(i=1, n, i*invphiNum(i)) - 1; \\ _Michel Marcus_, Jul 14 2025; using _Max Alekseyev_'s invphi.gp
%Y A385974 Cf. A000010, A005728, A014197.
%K A385974 nonn
%O A385974 1,2
%A A385974 _Hal M. Switkay_, Jul 13 2025