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

2, 3, 5, 7, 11, 13, 19, 23, 29, 43, 47, 59, 73, 97, 103, 151, 173, 181, 271, 397, 433, 491, 883, 941, 1087, 1103, 1163, 1193, 1229, 1427, 1471, 1697, 2143, 2273, 2657, 2903, 3533, 3677, 4073, 4129, 4201, 4259, 4637, 5023, 5107, 5953, 6163, 6599, 7177, 7237

Offset: 1

Author

Cino Hilliard, Nov 21 2002

Keywords

• easy
• nonn

Comments

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]

Examples

			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.
		

References

• H. Rademacher, Lectures on Elementary Number Theory, 1964. pp. 5-11.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Richard K. Guy, The Strong Law of Small Numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712.
• Eric Weisstein's World of Mathematics, Farey Sequence.

Crossrefs

Cf. A000010, A015614, A067282, A078334. [Jonathan Vos Post, Sep 06 2010]

Programs

Mathematica

fc[n_] := 1+Sum[EulerPhi[k], {k, 1, n}]; Select[fc/@Range[200], PrimeQ]

Extensions

Offset corrected by Amiram Eldar, Mar 01 2020