A240877 -id:A240877 - cp's OEIS frontend

A006843 Triangle read by rows: row n gives denominators of Farey series of order n.

1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 3, 2, 3, 4, 1, 1, 5, 4, 3, 5, 2, 5, 3, 4, 5, 1, 1, 6, 5, 4, 3, 5, 2, 5, 3, 4, 5, 6, 1, 1, 7, 6, 5, 4, 7, 3, 5, 7, 2, 7, 5, 3, 7, 4, 5, 6, 7, 1, 1, 8, 7, 6, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 6, 7, 8, 1, 1, 9, 8, 7, 6, 5, 9, 4, 7, 3, 8, 5, 7, 9, 2, 9, 7, 5, 8, 3, 7, 4, 9, 5, 6, 7, 8, 9, 1

Offset: 1

N. J. A. Sloane

			0/1, 1/1;
0/1, 1/2, 1/1;
0/1, 1/3, 1/2, 2/3, 1/1;
0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1;
0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1;
... = A006842/A006843.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152.
Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 199.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
A. O. Matveev, Farey Sequences, De Gruyter, 2017.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10563
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Andrey O. Matveev, Neighboring Fractions in Farey Subsequences, arXiv:0801.1981 [math.NT], 2008-2010.
Andrey O. Matveev, Farey Sequences: Errata + Haskell code
N. J. A. Sloane, Stern-Brocot or Farey Tree
Eric Weisstein's World of Mathematics, Farey Sequence.
Index entries for sequences related to Stern's sequences

Row n has A005728(n) terms. - Michel Marcus, Jun 27 2014
Row sums give A240877.
Cf. A006842 (numerators), A049455, A049456, A007305, A007306.
See also A177405/A177407.

Farey := proc(n) sort(convert(`union`({0},{seq(seq(m/k,m=1..k),k=1..n)}),list)) end: seq(denom(Farey(i)),i=1..5); # Peter Luschny, Apr 28 2009

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Flatten[ Table[ Denominator[ Farey[n]], {n, 9}]] (* Robert G. Wilson v, Apr 08 2004 *)
Table[Denominator[FareySequence[n]],{n,10}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 04 2016 *)

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(denominator(vf[i]), ", "));} \\ Michel Marcus, Jun 27 2014

More terms from Robert G. Wilson v, Apr 08 2004

Changed offset (=order of first row) to 1 by R. J. Mathar, Apr 26 2009

A213544 Sum of numerators of Farey Sequence of order n.

Original entry on oeis.org

1, 2, 5, 9, 19, 25, 46, 62, 89, 109, 164, 188, 266, 308, 368, 432, 568, 622, 793, 873, 999, 1109, 1362, 1458, 1708, 1864, 2107, 2275, 2681, 2801, 3266, 3522, 3852, 4124, 4544, 4760, 5426, 5768, 6236, 6556, 7376, 7628, 8531, 8971, 9511, 10017, 11098, 11482

Offset: 1

Anunay Kulshrestha, Jun 14 2012

			For n = 3, the Farey Sequence is 0/1, 1/3, 1/2, 2/3, 1/1. Thus a(3) = 0 + 1 + 1 + 2 + 1 = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Wikipedia, Farey Sequence

Similar to A133404 and A191607.
Partial sums of A023896.
Cf. A006842, A006843, A011755, A240877.

with(numtheory):
b:= n-> `if`(n=1, 1, n*phi(n)/2):
a:= proc(n) option remember; b(n) +`if`(n>1, a(n-1), 0) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 14 2012

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Table[ Total[ Numerator[ Farey[ n]]], {n, 0, 53}] (* Robert G. Wilson v, Apr 15 2014 *)
a[n_] := Sum[If[CoprimeQ[j, k], j, 0], {k, 1, n}, {j, 1, k}]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Dec 29 2014 *)
Table[Total[Numerator[FareySequence[n]]],{n,50}] (* Harvey P. Dale, Apr 21 2025 *)

a(n) = Sum_{k=1..n} A023896(k).
a(n) = A240877(n)/2. - Robert G. Wilson v, Apr 15 2014
a(n) ~ n^3/Pi^2 - Jean-François Alcover, Dec 29 2014
a(n) = (A011755(n)+1)/2. - Chai Wah Wu, Apr 04 2022

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;

A278051 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).

Original entry on oeis.org

2, 3, 10, 35, 252, 2772, 6435, 858, 680680, 12932920, 5290740, 121687020, 1029659400, 3088978200, 582272390700, 18050444111700, 128701918800, 25740383760, 70301729698200, 10043104242600, 109530094869795600, 523310453266801200, 51193413906534900, 481218090721428060

Offset: 1

N. J. A. Sloane, Nov 23 2016

			The fractions b(n) are 1/2, 2/3, 9/10, 38/35, 347/252, 4189/2772, 11767/6435, 1733/858, 1548081/680680, 31464371/12932920, 14680543/5290740, 353517989/121687020, 3350216417/1029659400, 10571768267/3088978200, ...

J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arithmetica 15.2 (1969): 181-187. See Eq. (20).

Cf. A006843, A005728, A240877, A278046, A278050.

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 30 do
t1:=denom(Farey(n));
t2:=add( 1/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
ans:=[op(ans),t2];
od:
ans;
map(numer,ans); # A278050
map(denom,ans); # A278051

A278047 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(kk'(k+k')), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

Original entry on oeis.org

1, 1, 7, 4, 37, 53, 707, 85, 179077, 289613, 379721, 641671, 62836087, 35819033, 6367281023, 55181728027, 13442946373, 490167893, 596530310479, 576997238399, 116144361532321, 4931206160615, 164890340129357, 1514840590670747, 10181612956306486603, 3295813969039399097

Offset: 1

N. J. A. Sloane, Nov 22 2016

			The fractions b(n) are 1/2, 1/3, 7/30, 4/21, 37/252, 53/396, 707/6435, 85/858, 179077/2042040, 289613/3527160, 379721/5290740, 641671/9360540, 62836087/1029659400, 35819033/617795640, ...

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

Cf. A006843, A005728, A240877, A278046, A278048.

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( 1/(t1[i]*t1[i+1]*(t1[i]+t1[i+1])),i=1..nops(t1)-1);
ans:=[op(ans),t2];
od:
ans;
map(numer,ans); # A278047
map(denom,ans); # A278048

A278048 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(kk'(k+k')), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).

Original entry on oeis.org

2, 3, 30, 21, 252, 396, 6435, 858, 2042040, 3527160, 5290740, 9360540, 1029659400, 617795640, 116454478140, 1061790830100, 283144221360, 10644519600, 14060345939640, 14060345939640, 2960272834318800, 130015019445168, 4653946718775900, 43747099156493460

Offset: 1

N. J. A. Sloane, Nov 22 2016

			The fractions b(n) are 1/2, 1/3, 7/30, 4/21, 37/252, 53/396, 707/6435, 85/858, 179077/2042040, 289613/3527160, 379721/5290740, 641671/9360540, 62836087/1029659400, 35819033/617795640, ...

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

Cf. A006843, A005728, A240877, A278046, A278047.

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( 1/(t1[i]*t1[i+1]*(t1[i]+t1[i+1])),i=1..nops(t1)-1);
ans:=[op(ans),t2];
od:
ans;
map(numer,ans); # A278047
map(denom,ans); # A278048

A278050 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

Original entry on oeis.org

1, 2, 9, 38, 347, 4189, 11767, 1733, 1548081, 31464371, 14680543, 353517989, 3350216417, 10571768267, 2114915577977, 69039991480573, 538250871701, 110983833313, 328448743696081, 48484885139543, 553270527392631611, 2736415713954900433, 286367762285513933, 2754025786313797907

Offset: 1

N. J. A. Sloane, Nov 23 2016

			The fractions b(n) are 1/2, 2/3, 9/10, 38/35, 347/252, 4189/2772, 11767/6435, 1733/858, 1548081/680680, 31464371/12932920, 14680543/5290740, 353517989/121687020, 3350216417/1029659400, 10571768267/3088978200, ...

J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arithmetica 15.2 (1969): 181-187. See Eq. (20).

Cf. A006843, A005728, A240877, A278046, A278051.

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 30 do
t1:=denom(Farey(n));
t2:=add( 1/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
ans:=[op(ans),t2];
od:
ans;
map(numer,ans); # A278050
map(denom,ans); # A278051

A278052 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum k*k'/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

Original entry on oeis.org

1, 4, 39, 52, 4069, 8573, 258017, 46639, 53371999, 113518551, 768140741, 1560819091, 242830653007, 169134016817, 38186305937387, 408881289764107, 143220706672837, 41293923006131, 9928250098118791, 10936700271572951, 97615258031147892517, 643700119549549507, 62211198375587838727

Offset: 1

N. J. A. Sloane, Nov 23 2016

			The fractions b(n) are 1/2, 4/3, 39/10, 52/7, 4069/252, 8573/396, 258017/6435, 46639/858, 53371999/680680, 113518551/1175720, 768140741/5290740, 1560819091/9360540, 242830653007/1029659400, 169134016817/617795640, 38186305937387/116454478140, ...

J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arithmetica 15.2 (1969): 181-187. See Eq. (21).

Cf. A006843, A005728, A240877, A278046-A278051, A278561 (denominators).

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 30 do
t1:=denom(Farey(n));
t2:=add( t1[i]*t1[i+1]/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
od:
ans;
map(numer,ans); # A278052
map(denom,ans); # A278561

A278561 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum k*k'/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).

Original entry on oeis.org

2, 3, 10, 7, 252, 396, 6435, 858, 680680, 1175720, 5290740, 9360540, 1029659400, 617795640, 116454478140, 1061790830100, 283144221360, 74511637200, 14060345939640, 14060345939640, 109530094869795600, 650075097225840, 51193413906534900, 481218090721428060

Offset: 1

N. J. A. Sloane, Nov 23 2016

			The fractions b(n) are 1/2, 4/3, 39/10, 52/7, 4069/252, 8573/396, 258017/6435, 46639/858, 53371999/680680, 113518551/1175720, 768140741/5290740, 1560819091/9360540, 242830653007/1029659400, 169134016817/617795640, 38186305937387/116454478140, ...

J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arithmetica 15.2 (1969): 181-187. See Eq. (21).

Cf. A006843, A005728, A240877, A278046-A278051, A278052 (numerators).

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 30 do
t1:=denom(Farey(n));
t2:=add( t1[i]*t1[i+1]/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
od:
ans;
map(numer,ans); # A278052
map(denom,ans); # A278561

cp's OEIS Frontend

A006843 Triangle read by rows: row n gives denominators of Farey series of order n.

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A213544 Sum of numerators of Farey Sequence of order n.

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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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A278051 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).

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A278047 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k*k'*(k+k')), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

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A278048 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k*k'*(k+k')), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).

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Maple

A278050 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

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A278052 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum k*k'/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

Views

Author

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Examples

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Programs

Maple

A278047 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(kk'(k+k')), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

A278048 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(kk'(k+k')), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).