A005728 -id:A005728 - cp's OEIS frontend

A049812 a(n)=number of Farey fractions of order n that are <=1/8; cf. A049805.

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 25, 27, 29, 31, 34, 35, 38, 40, 43, 45, 49, 50, 54, 56, 59, 61, 66, 68, 73, 76, 79, 82, 87, 89, 95, 97, 101, 104, 110, 112, 117, 120, 125, 129, 136, 138

Offset: 1

Clark Kimberling

nonn

Cf. A005728, A049806, A049807, A049808, A049809, A049810, A049811, A049813, A049814.

Farey[n_] := Union[Flatten[Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; f[n_] := Length@ Select[ Farey@ n, # <= 1/8 &]; Array[f, 60] (* Robert G. Wilson v, Nov 14 2012 *)

A049813 a(n)=number of Farey fractions of order n that are <=1/9; cf. A049805.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 27, 30, 31, 34, 36, 38, 40, 43, 44, 48, 50, 53, 55, 59, 60, 64, 66, 69, 72, 77, 79, 84, 86, 90, 93, 98, 100, 105, 108, 112, 115, 121, 122

Offset: 1

Clark Kimberling

nonn

Cf. A005728, A049806, A049807, A049808, A049809, A049810, A049811, A049812, A049814.

Farey[n_] := Union[Flatten[Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; f[n_] := Length@ Select[ Farey@ n, # <= 1/9 &]; Array[f, 60] (* Robert G. Wilson v, Nov 14 2012 *)

A049814 a(n)=number of Farey fractions of order n that are <=1/10; cf. A049805.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 30, 32, 34, 36, 39, 40, 43, 45, 47, 49, 53, 54, 58, 60, 63, 65, 69, 70, 74, 76, 80, 83, 88, 90, 94, 97, 101, 104, 109, 110

Offset: 1

Clark Kimberling

nonn

Cf. A005728, A049806, A049807, A049808, A049809, A049810, A049811, A049812, A049813.

Farey[n_] := Union[Flatten[Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; f[n_] := Length@ Select[ Farey@ n, # <= 1/10 &]; Array[f, 60] (* Robert G. Wilson v, Nov 14 2012 *)

A359694 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

Original entry on oeis.org

2, 10, 2, 70, 24, 218, 160, 4, 1254, 1068, 148, 16, 2254, 2414, 252, 26, 10082, 11760, 1980, 266, 12, 21410, 25958, 5096, 648, 36, 4, 53422, 68208, 14360, 1980, 168, 20, 86986, 118922, 24028, 3056, 248, 12, 0, 2, 255678, 346676, 84344, 12774, 1132, 110, 4, 2, 365674, 493530, 119820, 18600, 1624, 112, 4

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 11 2023

The number of vertices along each edge is A005728(n). No formula is known.

See A359692 for other images of the graph.

			The table begins:
2;
10, 2;
70, 24;
218, 160, 4;
1254, 1068, 148, 16;
2254, 2414, 252, 26;
10082, 11760, 1980, 266, 12;
21410, 25958, 5096, 648, 36, 4;
53422, 68208, 14360, 1980, 168, 20;
86986, 118922, 24028, 3056, 248, 12, 0, 2;
255678, 346676, 84344, 12774, 1132, 110, 4, 2;
365674, 493530, 119820, 18600, 1624, 112, 4;
917478, 1244492, 334096, 57080, 5700, 478, 16, 4;
1335398, 1862666, 495536, 82642, 8096, 676, 24, 6;
2107042, 2989864, 788340, 128378, 12536, 932, 52, 4;
3195474, 4557430, 1230300, 205352, 20516, 1664, 80, 4;
.
.

Scott R. Shannon, Image for n = 7.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Wikipedia, Farey sequence.

Cf. A359690 (vertices), A359691 (crossings), A359692 (regions), A359693 (edges), A005728, A290131, A359653, A358886, A358882, A006842, A006843.

Sum of row n = A359692(n).

A360042 Number of vertices in a Farey fan of order n.

Original entry on oeis.org

4, 6, 11, 17, 29, 39, 59, 79, 107, 133, 175, 213, 271, 323, 385, 451, 541, 621, 731, 835, 955, 1073, 1225, 1367, 1541, 1707, 1897, 2087, 2321, 2535, 2801, 3061, 3345, 3625, 3937, 4243, 4609, 4957, 5335, 5713, 6155, 6569, 7055, 7529, 8031, 8531, 9101, 9649, 10265, 10859

Offset: 1

Scott R. Shannon, N. J. A. Sloane and M. Douglas McIlroy Jan 23 2023

nonn

See the reference for the definition of a 'Farey fan'.
The number of vertices along each edge is A005728(n), while the number of regions is conjectured to equal A005598(n) = 1 + Sum_{i=1..n} (n-i+1)*phi(i). The regions count the number of distinct approximate representations of straight lines y = mx + b that can be drawn on an x-y integer raster, where x, y, and b are restricted to [0,n) and 0 <= m <=1.
It is also worth noting that for 3 <= n <= 10 this sequence equals 2*A005728(n) + A174030(n-2), where A174030(n) = Sum_{i=1..n} (i where phi(i)|i). That is, the number of internal vertices of the Farey fan equals A174030(n) in this range. This may suggest a possible attack on finding a formula for the present sequence.

M. Douglas McIlroy, A Note on Discrete Representation of Lines, AT&T Technical Journal, 64 (1985), 481-490.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 10.

Cf. A005598 (regions), A360043 (edges), A360044 (k-gons), A005728, A174030, A359974, A359968, A359690.

A359693 Number of edges in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

Original entry on oeis.org

6, 24, 162, 670, 4456, 8942, 44470, 98902, 259114, 438552, 1330566, 1897164, 4893752, 7246502, 11544278, 17678880

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 11 2023

The number of vertices along each edge is A005728(n). No formula for a(n) is known.

See A359690 and A359692 for images of the graph.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Wikipedia, Farey sequence.

Cf. A359690 (vertices), A359691 (crossings), A359692 (regions), A359694 (k-gons), A005728, A290132, A359655, A358888, A358884, A006842, A006843.

a(n) = A359690(n) + A359692(n) - 2*A005728(n) + 1 by Euler's formula.

A049643 Number of fractions in Farey series of order n.

Original entry on oeis.org

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

Clark Kimberling

Essentially the same as A005728.

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)

Cf. A000010.

[0] cat [n le 1 select 2 else Self(n-1)+EulerPhi(n): n in [1..60]]; // G. C. Greubel, Dec 06 2017

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 *)

for(n=0, 30, print1(if(n==0, 0, 1+sum(k=1, n, eulerphi(k))), ", ")) \\ G. C. Greubel, Dec 06 2017

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

A278046 Let v = list of denominators of Farey series of order n (see A006843); a(n) = sum of products of adjacent terms of v.

Original entry on oeis.org

1, 4, 18, 44, 124, 186, 424, 636, 1038, 1378, 2368, 2852, 4516, 5510, 7030, 8734, 12542, 14168, 19526, 22206, 26658, 30728, 40342, 44190, 54590, 61402, 72328, 80196, 99684, 105644, 129514, 143162, 161422, 176926, 201566, 214538, 255386, 277160, 307736, 329096, 384856, 402412, 466826, 499166

Offset: 1

N. J. A. Sloane, Nov 22 2016

nonn

Note that the sum of the reciprocals of these products is 1.

			When n = 4, v = [1,4,3,2,3,4,1], so a(4) = 1*4 + 4*3 + 3*2 + 2*3 + 3*4 + 4*1 = 44.

J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arithmetica 15.2 (1969): 181-187.

Cf. A006843, A005728, A240877.

Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
ans:=[];
for n from 1 to 50 do
t1:=denom(Farey(n));
t2:=add( t1[i]*t1[i+1],i=1..nops(t1)-1);
ans:=[op(ans),t2];
od:
ans;

A358884 The number of edges in a Farey diagram of order (n,n).

Original entry on oeis.org

8, 92, 816, 3276, 13040, 29452, 82128, 160656, 328212, 556040, 1065660, 1592368, 2768168, 4026972, 6083804, 8572272, 13075848, 17078512, 24932940, 32266036

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

See the linked references for further details.

The first diagram where not all edge points are connected is n = 3. For example a line connecting points (0,1/3) and (1/3,0) has equation 3*y - 6*x - 1 = 0, and as one of the x or y coefficients is greater than n (3 in this case) the line is not included.

Alain Daurat et al., About the frequencies of some patterns in digital planes. Application to area estimators. Computers & graphics. 33.1 (2009), 11-20.
Daniel Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures. Applicable Analysis and Discrete Mathematics. 9 (2015), 73-84.
Wikipedia, Farey sequence.

Cf. A358882 (regions), A358883 (vertices), A358885 (k-gons), A006842, A006843, A005728, A358888.
See A358298 for definition of Farey diagram Farey(m,n).
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

a(n) = A358882(n) + A358883(n) - 1 by Euler's formula.

A359653 Number of regions formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

Original entry on oeis.org

1, 4, 96, 728, 7840, 17744, 104136, 246108, 681704, 1187200, 3719496, 5396692, 14149896

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 09 2023

The number of points internal to each edge is given by A005728(n) - 2.

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.

Cf. A359654 (vertices), A359655 (edges), A359656 (k-gons), A005728, A358886, A358882, A355798, A358948, A006842, A006843.

a(n) = A359655(n) - A359654(n) + 1 by Euler's formula.

cp's OEIS Frontend

A049812 a(n)=number of Farey fractions of order n that are <=1/8; cf. A049805.

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A049813 a(n)=number of Farey fractions of order n that are <=1/9; cf. A049805.

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Mathematica

A049814 a(n)=number of Farey fractions of order n that are <=1/10; cf. A049805.

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Mathematica

A359694 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

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A360042 Number of vertices in a Farey fan of order n.

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A359693 Number of edges in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

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A049643 Number of fractions in Farey series of order n.

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Magma

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PARI

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A278046 Let v = list of denominators of Farey series of order n (see A006843); a(n) = sum of products of adjacent terms of v.

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Maple

A358884 The number of edges in a Farey diagram of order (n,n).

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A359653 Number of regions formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

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