A006843 -id:A006843 - cp's OEIS frontend

A324796 Irregular triangle read by rows: row n gives numerators of fractions in the Farey subsequence B(m).

0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 0, 1, 1, 1, 2, 3, 1, 4, 3, 2, 3, 4, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 1, 5, 4, 3, 5, 2, 5, 3, 4, 5, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 1, 6, 5, 4, 3, 5, 2, 5, 3, 4, 5, 6, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 5, 3, 4, 5, 6, 1, 7, 6, 5, 4, 7, 3, 5, 7, 2, 7, 5, 3, 7, 4, 5, 6, 7, 1

Offset: 1

N. J. A. Sloane, Sep 10 2019

B(n) is denoted by F(B(2n),n) in Matveev (2017) - see definition on page 1. B(n) consists of the terms h/k of the Farey series F_{2n} such that k-n <= h <= n.

A049691 gives the row lengths.

			The first few sequences B(1), B(2), B(3), B(4) are:
[0, 1/2, 1],
[0, 1/3, 1/2, 2/3, 1],
[0, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1],
[0, 1/5, 1/4, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 3/4, 4/5, 1], [0, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 4/9, 1/2, 5/9, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 1],
...

A. O. Matveev, Farey Sequences, De Gruyter, 2017.

A. O. Matveev, Farey Sequences: Errata + Haskell code

Cf. A006842/A006843, A049691, A324797 (denominators).

Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
B := proc(m) local a,i,h,k; global Farey; a:=[];
for i in Farey(2*m) do
h:=numer(i); k:=denom(i);
if (h <= m) and (k-m <= h) then a:=[op(a),i]; fi; od: a; end;

A324797 Irregular triangle read by rows: row n gives denominators of fractions in the Farey subsequence B(m).

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 3, 5, 2, 5, 3, 4, 1, 1, 5, 4, 3, 5, 7, 2, 7, 5, 3, 4, 5, 1, 1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2, 9, 7, 5, 8, 3, 7, 4, 5, 6, 1, 1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2, 11, 9, 7, 5, 8, 3, 7, 4, 5, 6, 7, 1, 1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2, 13, 11, 9, 7, 12, 5, 8, 11, 3, 10, 7, 4, 9, 5, 6, 7, 8, 1

Offset: 1

N. J. A. Sloane, Sep 10 2019

B(n) is denoted by F(B(2n),n) in Matveev (2017) - see definition on page 1. B(n) consists of the terms h/k of the Farey series F_{2n} such that k-n <= h <= n.

A049691 gives the row lengths.

			The first few sequences B(1), B(2), B(3), B(4) are:
[0, 1/2, 1],
[0, 1/3, 1/2, 2/3, 1],
[0, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1],
[0, 1/5, 1/4, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 3/4, 4/5, 1], [0, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 4/9, 1/2, 5/9, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 1],
...

A. O. Matveev, Farey Sequences, De Gruyter, 2017.

A. O. Matveev, Farey Sequences: Errata + Haskell code

Cf. A006842/A006843, A049691, A324796 (numerators).

Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
B := proc(m) local a,i,h,k; global Farey; a:=[];
for i in Farey(2*m) do
h:=numer(i); k:=denom(i);
if (h <= m) and (k-m <= h) then a:=[op(a),i]; fi; od: a; end;

A376832 Irregular triangle read by rows: the n-th row gives the number of points of an n X n square lattice that lie above or to the left of a line of increasing slope that passes through two lattice points one of which is the bottom-left corner of the lattice, (0, 0).

Original entry on oeis.org

2, 1, 0, 6, 5, 3, 2, 0, 12, 11, 10, 9, 6, 5, 4, 3, 0, 20, 19, 18, 17, 16, 15, 14, 10, 9, 8, 7, 6, 5, 4, 0, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 0, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 0

Offset: 2

Stefano Spezia, Dec 22 2024

The increasing slopes of the line are given by the Farey series of order n - 1. Specifically, they are given by the fractions A006842(n-1)/A006843(n-1) followed by their reciprocals A006843(n-1)/A006842(n-1) in reverse order, with the fraction 1/1 included only once.

			The irregular triangle begins as:
   2,  1,  0;
   6,  5,  3,  2,  0;
  12, 11, 10,  9,  6,  5,  4,  3, 0;
  20, 19, 18, 17, 16, 15, 14, 10, 9, 8, 7, 6, 5, 4, 0;
  ...

Stefano Spezia, Table of n, a(n) for n = 2..9081 (first 30 rows of the irregular triangle)

Cf. A002378, A006842, A006843, A118403 (row lengths), A161680, A379540 (row sums).

A118403[n_]:=SeriesCoefficient[(1-2*x+2*x^2)*(1+x^2)/(1-x)^3,{x,0,n}]; T[n_,k_]:=If[1<=k<(A118403[n]+1)/2,n(n-1)-k+1,If[(A118403[n]+1)/2<=k<A118403[n],n(n-1)/2-k+(A118403[n]+1)/2,0]]; Table[T[n,k],{n,2,7},{k,A118403[n]}]//Flatten

T(n, k) = n*(n - 1) - k + 1 for 1 <= k < (A118403(n)+1)/2.
T(n, k) = n*(n - 1)/2 - k + (A118403(n)+1)/2 for (A118403(n)+1)/2 <= k < A118403(n).
T(n, A118403(n)) = 0.

A061560 Lengths of Farey-series (A005728) which are not primes.

Original entry on oeis.org

1, 33, 65, 81, 121, 129, 141, 201, 213, 231, 243, 279, 309, 325, 345, 361, 385, 451, 475, 531, 543, 585, 605, 629, 651, 697, 713, 755, 775, 807, 831, 901, 965, 1001, 1029, 1261, 1309, 1329, 1395, 1495, 1565, 1589, 1661, 1737, 1773, 1833, 1857, 1935, 1967

Offset: 1

Frank Ellermann, May 17 2001

nonn

			a(3) = 65 = 5*13 is not a prime, A055197(3-1) = 14, A005728(14) = 65.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005728, A055197, A006842, A006843, A002088, A000010, A055201.

seq = {1}; sum = 1; Do[sum += EulerPhi[k]; If[!PrimeQ[sum], AppendTo[seq, sum]], {k, 1, 80}]; seq (* Amiram Eldar, Mar 01 2020 *)

a(n) = A005728(A055197(n-1)) for n > 1, a(1) = A005728(0).

More terms from Vladeta Jovovic, Jun 03 2001

Offset corrected by Amiram Eldar, Mar 01 2020

A278148 Triangle T(n, m) giving in row n the numerators of the fractions for the Farey dissection of order n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 1, 2, 2, 3, 5, 4, 1, 2, 2, 3, 3, 4, 5, 5, 7, 5, 1, 2, 2, 2, 3, 3, 4, 5, 5, 7, 9, 6, 1, 2, 2, 2, 3, 3, 3, 5, 4, 5, 7, 5, 7, 8, 7, 9, 11, 7, 1, 2, 2, 2, 2, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 7, 8, 7, 9, 11, 13, 8, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 7, 5, 6, 9, 7, 8, 7, 7, 8, 10, 11, 9, 11, 13, 15, 9

Offset: 1

Wolfdieter Lang, Nov 22 2016

For the denominators see A278149.
The length of row n is A002088(n) = A005728(n) - 1.
In the Hardy reference one finds from the Farey fractions of order n >= 2 (see A006842/A006843) a dissection of the interval [1/(n+1), n/(n+1)] into A015614(n) = A005728(n) - 2  intervals J(n,j) = [l(n,j), r(n,j)], j = 1..A015614(n). They are obtained from three consecutive Farey fractions of order n: p(n,j-1)/q(n,j-1), p(n,j)/q(n,j), p(n,j+1)/q(n,j+1) by l(n,j) = p(n,j)/q(n,j) - 1/(q(n,j)*(q(n,j) + q(n,j-1))) = (p(n,j) + p(n,j-1))/(q(n,j) + q(n,j-1)) and r(n,j) = p(n,j)/q(n,j) + 1/(q(n,j)*(q(n,j) + q(n,j+1))) = (p(n,j) + p(n,j+1))/(q(n,j) + q(n,j+1)). (Hardy uses N for n, p/q - Chi_{p,q}'' for l (left) and p/q + Chi_{p,q}' for r (right)). For the second equations in l(n,j) and r(n,j) see the identities in Hardy-Wright, p. 23, Theorem 28.
Due to r(n,j) = l(n,j+1), for n >= 2 and j=1..A015614(n), it is sufficient to give for this Farey dissection of order n >= 2 only the two endpoints of the interval [1/(n+1), n/(n+1)] and the A015614(n) - 1 = A002088 - 2 = A005728(n) - 3 inner boundary points. The present table gives the numerators of these fractions. See the example section. For n = 1 we add the row 1/2 in accordance with A002088(1) = 1.

			The triangle T(n, m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11 12 ...
1:  1
2:  1 2
3:  1 2 3 3
4:  1 2 2 3 5 4
5:  1 2 2 3 3 4 5 5 7  5
6:  1 2 2 2 3 3 4 5 5  7  9  6
...
n = 7:  1 2 2 2 3 3 3 5 4 5 7 5 7 8 7 9 11 7,
n = 8:  1 2 2 2 2 3 3 4 5 5 4 5 7 8 7 7 8 7 9 11 13 8,
n = 9:  1 2 2 2 2 3 3 3 3 4 5 5 7 5 6 9 7 8 7 7 8 10 11 9 11 13 15 9,
n = 10: 1 2 2 2 2 2 3 3 3 5 4 4 5 5 7 5 6 9 7 8 7 9 12 8 10 11 9 11 13 15 17 10.
.............................................
The fractions T(n,m)/A278149(n, m) begin:
n\m 1    2   3   4   5   6   7   8   9  10
1: 1/2
2: 1/3  2/3
3: 1/4  2/5 3/5 3/4
4: 1/5  2/7 2/5 3/5 5/7 4/5
5: 1/6  2/9 2/7 3/8 3/7 4/7 5/8 5/7 7/9 5/6
...
n = 6: 1/7 2/11 2/9 2/7 3/8 3/7 4/7 5/8 5/7 7/9 9/11 6/7,
n = 7: 1/8 2/13 2/11 2/9 3/11 3/10 3/8 5/12 4/9 5/9 7/12 5/8 7/10 8/11 7/9 9/11 11/13 7/8,
n = 8: 1/9 2/15 2/13 2/11 2/9 3/11 3/10 4/11 5/13 5/12 4/9 5/9 7/12 8/13 7/11 7/10 8/11 7/9 9/11 11/13 13/15 8/9,
n = 9: 1/10 2/17 2/15 2/13 2/11 3/14 3/13 3/11 3/10 4/11 5/13 5/12 7/16 5/11 6/11 9/16 7/12 8/13 7/11 7/10 8/11 10/13 11/14 9/11 11/13 13/15 15/17 9/10,
n = 10: 1/11 2/19 2/17 2/15 2/13 2/11 3/14 3/13 3/11 5/17 4/13 4/11 5/13 5/12 7/16 5/11 6/11 9/16 7/12 8/13 7/11 9/13 12/17 8/11 10/13 11/14 9/11 11/13 13/15 15/17 17/19 10/11.
.............................................
For n = 5 the actual intervals J(5,j), j= 1..9 are then:
[1/6, 2/9], [2/9, 2/7], [2/7, 3/8], [3/8, 3/7], [3/7, 4/7], [4/7, 5/8], [5/8, 5/7], [5/7, 7/9], [7/9, 5/6].

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 23, 29 - 31.

Cf. A002088, A005728, A006842/A006843, A015614, A278149.

T(1, 1) = 1 and for n>= 2: T(n, 1) = 1, T(n, A002088(n)) = n and for m = 2..(A002088(n) - 1): T(n, m) = numerator(l(n,m)) = numerator( p(n,m)/q(n,m) - 1/(q(n,m)*(q(n,m) + q(n,m-1)))).

A122632 Table T(n,k) = number of initial segments of Beatty sequences for numbers > 1 of length k, cutting sequence so that all terms are < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 1, 3, 2, 2, 3, 1, 1, 4, 3, 2, 3, 4, 1, 1, 4, 3, 3, 3, 3, 4, 1, 1, 5, 3, 4, 2, 4, 3, 5, 1, 1, 5, 4, 3, 3, 3, 3, 4, 5, 1, 1, 6, 4, 4, 5, 2, 5, 4, 4, 6, 1, 1, 6, 4, 4, 4, 4, 4, 4, 4, 4, 6, 1, 1, 7, 5, 5, 4, 6, 2, 6, 4, 5, 5, 7, 1, 1, 7, 5, 5, 4, 6, 4, 4, 6, 4, 5, 5, 7, 1

Offset: 1

Franklin T. Adams-Watters, Oct 20 2006

Enumerate all rational numbers q in [0,1) with denominator <= n. T(n,k) is the number of these with floor(n*q) = k-1. Problem suggested by David W. Wilson.

			T(6,3) = 2; the sequences for n=6, k=3 are 0,2,4 and 0,2,5. The sequence 0,1,3 is not counted because the next term of a Beatty sequence beginning 0,1,3 must be 4 or 5, so 0,1,3 is not a Beatty sequence truncated to numbers less than 6.

Cf. A002088 (row sums), A006842/A006843 (Farey fractions).

Flatten@Table[Count[Select[Union@Flatten@Outer[Divide, Range[n + 1] - 1, Range[n]] , # <= 1 &], ?(Floor[n #] == k &)], {n, 12}, {k, n}] (* _Birkas Gyorgy *)

A244396 a(n) = Sum_{k=1, n} phi(k)*index(k, n), with phi(k) the Euler totient A000010(k) and index(k,n) the position of 1/k in the n-th row of the Farey sequence of order k, A049805(n,k).

Original entry on oeis.org

2, 5, 12, 21, 39, 54, 87, 117, 162, 204, 279, 333, 435, 516, 624, 732, 900, 1017, 1224, 1380, 1590, 1785, 2082, 2286, 2616, 2886, 3237, 3543, 4005, 4305, 4830, 5238, 5748, 6204, 6816, 7266, 8004, 8571, 9279, 9879, 10779, 11373, 12360, 13110, 14010, 14835

Offset: 1

Michel Marcus, Jun 27 2014

nonn

R. Tomás, From Farey sequences to resonance diagrams, Phys. Rev. ST Accel. Beams 17, 014001 - Published 29 January 2014.
R. Tomás, Asymptotic behavior of a series of Euler's Totient function times the cardinality of truncated Farey sequences, arXiv:1406.6991 [math.NT], 2014 (see Chapter 5, Evaluating ...).

Cf. A000010, A049805, A006842, A006843.

a[n_] := With[{f = FareySequence[n]}, Sum[EulerPhi[k] FirstPosition[f, 1/k ][[1]], {k, 1, n}]]; Array[a, 50] (* Jean-François Alcover, Sep 26 2018 *)

farey(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k););); vecsort(Set(vf));}
a(n) = my(row = farey(n)); sum(k=1, n, eulerphi(k)*vecsearch(row, 1/k));

a(n) = Sum_{k=1, n} A000010(k)*A049805(k, n).

a(n) = n^3/(6*zeta(3)) + O(n^2). (see (22) in Tomás link).

A246064 A variation of the Farey sequence read by rows; the row of order n is defined by the greatest whole numbers less than or equal to the values generated when the numerators of the Farey sequence (order n) are replaced by n itself.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 4, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 6, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 7, 8, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 8

Offset: 1

Neill L. Robson, Aug 24 2014

Consider the area on the coordinate plane swept by a polar angle ranging from 0 to Pi/4 radians. Take every integer coordinate pair from that area whose coordinate components do not exceed the magnitude of n (i.e., if n=9, coordinate (9, 3) is chosen but not (10, 3) ). There would be a distinct number of angles between 0 and Pi/4 radians representing each point's polar angle. For large enough n, some points would share the same angle; for instance, where n=9, Cartesian points (3, 1), (6, 2), and (9, 3) would be chosen, yet all would have the radian angle arctan(1/3).
Consider these unique angle values in ascending order. Replace each angle with the number of points that could be represented in polar form with that angle (i.e., arctan(1/3) from the above example would be replaced with 3). This will result in row n of the sequence.
Row n has A005728(n) terms.

			As an example, the 4th row of the sequence (n=4) is 4, 1, 1, 2, 1, 1, 4.
Triangle starts:
  1, 1;
  2, 1, 2;
  3, 1, 1, 1, 3;
  4, 1, 1, 2, 1, 1, 4;
  5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5;
  ...

Cf. A006843, A005728.

<. (% A006843)~ n
NB. A006843 is assumed to return a list of Farey denominators of order n, as described in the OEIS.

row(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k); ); ); vf = vecsort(Set(vf)); for (i=1, #vf, print1(n\denominator(vf[i]), ", ")); } \\ Michel Marcus, Sep 04 2014

Row n of the sequence (sequence of order n) is given by floor(n/A006843(n)).

A248832 The sum of denominators of unreduced mediants in Farey sequences of orders 1,2,..,n.

Original entry on oeis.org

2, 6, 16, 34, 72, 122, 214, 338, 516, 734, 1062, 1438, 1970, 2586, 3322, 4186, 5322, 6566, 8152, 9898, 11896, 14114, 16838, 20254, 23982, 28196, 32746, 38108, 43710, 50242, 57286, 64990, 73238, 82326, 91846, 102698, 114234, 126706, 139818

Offset: 0

Nehul Yadav, Oct 15 2014

			For n=4, the unreduced mediants are 1/2, 2/4, 5/10, 9/18, hence the sum of denominators is 2+4+10+18 = 34.

Wikipedia, Mediant.
Wikipedia, Farey Sequence.

Cf. A007306, A006843.

a(n) = 2*Sum_{k= 1,2,3,4..n} A213544(k).

A255541 a(n) = 1+Sum_{k=1..2^n-1} A000010(k).

Original entry on oeis.org

1, 2, 5, 19, 73, 309, 1229, 4959, 19821, 79597, 318453, 1274563, 5097973, 20397515, 81591147, 326371001, 1305482159, 5222040189, 20888133573, 83552798667, 334211074959, 1336845501841, 5347382348679, 21389531880435, 85558125961121, 342232529890275, 1368930120480617, 5475720508827645, 21902882035220391, 87611528574186091, 350446114129452131, 1401784457568941917, 5607137830212707769

Offset: 0

Robert G. Wilson v, Feb 24 2015

nonn

Number of fractions in Farey series of order 2^n-1.

			For each n, measure the size of the set of reduced fractions with a denominator less than 2^n:
a(0) = 1 since the set of reduced fractions with denominator less than 2^0 = 1 is {0}.
a(1) = 2 since the set of reduced fractions with denominator less than 2^1 = 2 is {0, 1}.
a(2) = 5 since the set of reduced fractions with denominator less than 2^2 = 4 is {0, 1/3, 1/2, 2/3, 1}.
a(3) = 19 since the set of reduced fractions with denominator less than 2^3 = 8 is {0, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1}.

D. T. Ashley, J. P. DeVoe, K. Perttunen, C. Pratt, and A. Zhigljavsky, On Best Rational Approximations Using Large Integers
Wikipedia, Farey Sequence

Cf. A007305, A007306, A000010, A049643, A006842/A006843 (Farey fractions).

k = s = 1; lst = {}; Do[While[k < 2^n, s = s + EulerPhi@ k; k++]; AppendTo[lst, s], {n, 0, 26}]; lst
a[n_] := 1 + (1/2) Sum[ MoebiusMu[k]*Floor[n/k]*Floor[1 + n/k], {k, n}]; Array[a, 27, 0]

a(n) ~ (2^n-1)^2 / Pi.
a(n) = 2+A015614(2^n-1).
a(n) = A005728 (2^n-1). - Michel Marcus, Feb 27 2015
a(n) = (3+A018805(2^n-1))/2. - Colin Linzer, Aug 06 2025

cp's OEIS Frontend

A324796 Irregular triangle read by rows: row n gives numerators of fractions in the Farey subsequence B(m).

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A324797 Irregular triangle read by rows: row n gives denominators of fractions in the Farey subsequence B(m).

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A376832 Irregular triangle read by rows: the n-th row gives the number of points of an n X n square lattice that lie above or to the left of a line of increasing slope that passes through two lattice points one of which is the bottom-left corner of the lattice, (0, 0).

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A061560 Lengths of Farey-series (A005728) which are not primes.

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A278148 Triangle T(n, m) giving in row n the numerators of the fractions for the Farey dissection of order n.

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A122632 Table T(n,k) = number of initial segments of Beatty sequences for numbers > 1 of length k, cutting sequence so that all terms are < n.

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A244396 a(n) = Sum_{k=1, n} phi(k)*index(k, n), with phi(k) the Euler totient A000010(k) and index(k,n) the position of 1/k in the n-th row of the Farey sequence of order k, A049805(n,k).

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A246064 A variation of the Farey sequence read by rows; the row of order n is defined by the greatest whole numbers less than or equal to the values generated when the numerators of the Farey sequence (order n) are replaced by n itself.

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