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A078334 Primes in A005728, which counts the terms in the Farey sequence of order n.

%I A078334 #25 Feb 16 2025 08:32:48
%S A078334 2,3,5,7,11,13,19,23,29,43,47,59,73,97,103,151,173,181,271,397,433,
%T A078334 491,883,941,1087,1103,1163,1193,1229,1427,1471,1697,2143,2273,2657,
%U A078334 2903,3533,3677,4073,4129,4201,4259,4637,5023,5107,5953,6163,6599,7177,7237
%N A078334 Primes in A005728, which counts the terms in the Farey sequence of order n.
%C A078334 Guy, in his Example 8, citing Leo Moser as his source, noted that the first 9 values of A005728(n) = 1 + Sum_{i=1..n} phi(i) = 1 + Sum_{i=1..n} A000010(i) are all primes, but that the pattern breaks down at A005728(10) = 33 = 3*11. As Guy warns, in several paraphrases of the same law, "Capricious coincidences cause careless conjectures." That is, for 1 <= n <= 9 we have A005728(n) = A078334(n), but for n > 9 we sometimes (n = {11, 12, 13, 15, 17, 18, 22, ...}) have A005728(n) prime, but other times (n = {10, 14, 16, 19, 20, 21, ...}) have A005728(n) composite. [_Jonathan Vos Post_, Sep 06 2010]
%D A078334 H. Rademacher, Lectures on Elementary Number Theory, 1964. pp. 5-11.
%H A078334 Amiram Eldar, <a href="/A078334/b078334.txt">Table of n, a(n) for n = 1..10000</a>
%H A078334 Richard K. Guy, <a href="http://www.jstor.org/stable/2322249">The Strong Law of Small Numbers</a>, Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A078334 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FareySequence.html">Farey Sequence.</a>
%e A078334 The Farey sequence of order 6 is {0, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1}, which has 13 terms, so 13 is in the sequence.
%t A078334 fc[n_] := 1+Sum[EulerPhi[k], {k, 1, n}]; Select[fc/@Range[200], PrimeQ]
%Y A078334 Cf. A000010, A015614, A067282, A078334. [_Jonathan Vos Post_, Sep 06 2010]
%K A078334 easy,nonn
%O A078334 1,1
%A A078334 _Cino Hilliard_, Nov 21 2002
%E A078334 Offset corrected by _Amiram Eldar_, Mar 01 2020