A049643 Number of fractions in Farey series of order n.

0, 2, 3, 5, 7, 11, 13, 19, 23, 29, 33, 43, 47, 59, 65, 73, 81, 97, 103, 121, 129, 141, 151, 173, 181, 201, 213, 231, 243, 271, 279, 309, 325, 345, 361, 385, 397, 433, 451, 475, 491, 531, 543, 585, 605, 629, 651, 697, 713, 755, 775, 807, 831, 883

Offset

0,2

Comments

Essentially the same as A005728.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Formula

a(n) = A049641(2*n).

From G. C. Greubel, Dec 13 2017: (Start)

a(n) = 1 + Sum_{k=1..n} phi(k), with a(0)=0.

a(n) = A005728(n) for n >= 1. (End)

a(n) = a(n-1) + phi(n) for n > 1. - Robert G. Wilson v, Dec 13 2017

Mathematica

a[0] = 0; a[n_] := 1 + Sum[EulerPhi[k], {k, 1, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 27 2015 *)
a[0] = 0; a[1] = 2; a[n_] := a[n -1] + EulerPhi[n]; Array[a, 55, 0] (* Robert G. Wilson v, Dec 13 2017 *)
Join[{0}, Rest[Accumulate[EulerPhi[Range[0, 60]]]+1]] (* Harvey P. Dale, Oct 16 2018 *)
a[n_] := If[n == 0, 0, FareySequence[n] // Length];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 16 2022 *)

Prog

(PARI) for(n=0, 30, print1(if(n==0, 0, 1+sum(k=1, n, eulerphi(k))), ", ")) \\ G. C. Greubel, Dec 06 2017
(Magma) [0] cat [n le 1 select 2 else Self(n-1)+EulerPhi(n): n in [1..60]]; // G. C. Greubel, Dec 06 2017

Crossrefs

Cf. A000010.

Sequence in context: A152900 A079151 A274335 * A005728 A050437 A096246

Adjacent sequences: A049640 A049641 A049642 * A049644 A049645 A049646

Keyword

nonn,easy,nice

Author

Clark Kimberling

Status

approved