A226314 -id:A226314 - cp's OEIS frontend

A243808 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1 and x+y odd.

2, 5, 7, 9, 12, 14, 16, 20, 23, 25, 27, 29, 31, 33, 35, 38, 40, 44, 46, 48, 52, 54, 57, 59, 61, 63, 65, 67, 71, 73, 77, 80, 82, 84, 86, 88, 90, 92, 94, 96, 100, 102, 104, 107, 109, 113, 119, 121, 123, 125, 127, 129, 131, 133, 135, 138, 140

Offset: 1

Frank M Jackson, Jun 13 2014

nonn

If {x,y} are used as the generators of primitive Pythagorean triples (a,b,c) where a=y^2-x^2, b=2x*y and c=x^2+y^2, then the above sequence enumerates all PPT's as a 1-to-1 mapping into the integers.

			.  j       {A226314(n),A054531(n)}, 1<=i<=j<=12 and n=i+j(j-1)/2
.  --   ------------------------------------------------------------
.   1:  1,1
.   2:  1,2 2,1
.   3:  1,3 2,3 3,1
.   4:  1,4 3,2 3,4 4,1
.   5:  1,5 2,5 3,5 4,5 5,1
.   6:  1,6 4,3 5,2 5,3 5,6 6,1
.   7:  1,7 2,7 3,7 4,7 5,7 6,7 7,1
.   8:  1,8 5,4 3,8 7,2 5,8 7,4 7,8 8,1
.   9:  1,9 2,9 7,3 4,9 5,9 8,3 7,9 8,9 9,1
.  10:  1,10 6,5 3,10 7,5 9,2 8,5 7,10 9,5 9,10 10,1
.  11:  1,11 2,11 3,11 4,11 5,11 6,11 7,11 8,11 9,11 10,11 11,1
.  12:  1,12 7,6 9,4 10,3 5,12 11,2 7,12 11,3 11,4 11,6 11,12 12,1 .
a(4)=9, as the 4th PPT is generated from the 9th term of the triangular array at index (3,4). This gives (x,y) as (3,4) and it generates the PPT (7,24,25). Conversely the PPT (7,24,25) gives (x,y) = (sqrt((25-7)/2), sqrt((25+7)/2)) = (3,4). It is the 9th term of the triangular array and the 4th term of the enumerating sequence.

Frank M Jackson, Table of n, a(n) for n = 1..57
Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A054531, A169581, A226314.

ratmap[p_, q_] := (q(q-1)/2+p); mm=20; lst={}; Do[If[OddQ[m+n]&&GCD[m, n]==1, AppendTo[lst, n/m]], {m, 1, mm}, {n, 1, m}]; Sort@Table[ratmap[Numerator[lst[[k]]], Denominator[lst[[k]]]], {k, 1, Length[lst]}]

A226555 Numerators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n.

Original entry on oeis.org

1, 5, 4, 13, 7, 25, 10, 33, 17, 45, 16, 69, 19, 65, 38, 81, 25, 109, 28, 125, 55, 105, 34, 177, 53, 125, 68, 181, 43, 241, 46, 193, 89, 165, 100, 301, 55, 185, 106, 321, 61, 349, 64, 293, 167, 225, 70, 433, 109, 341, 140, 349, 79, 433, 162, 465, 157, 285, 88

Offset: 1

Reinhard Zumkeller, Jun 10 2013

			.   n         A226314(n,k) / A054531(n,k), 1<=k<=n<=12           row sums
.  --   -------------------------------------------------------- --------
.   1:  1                                                           1
.   2:  1/2 2                                                       5/2
.   3:  1/3 2/3 3                                                   4
.   4:  1/4 3/2 3/4 4                                              13/2
.   5:  1/5 2/5 3/5 4/5 5                                           7
.   6:  1/6 4/3 5/2 5/3 5/6 6                                      25/2
.   7:  1/7 2/7 3/7 4/7 5/7 6/7 7                                  10
.   8:  1/8 5/4 3/8 7/2 5/8 7/4 7/8 8                              33/2
.   9:  1/9 2/9 7/3 4/9 5/9 8/3 7/9 8/9 9                          17
.  10:  1/10 6/5 3/10 7/5 9/2 8/5 7/10 9/5 9/10 10                 45/2
.  11:  1/11 2/11 3/11 4/11 5/11 6/11 7/11 8/11 9/11 10/11 11      16
.  12:  1/12 7/6 9/4 10/3 5/12 11/2 7/12 11/3 11/4 11/6 11/12 12   69/2 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A040001 (denominators).

import Data.Ratio ((%), numerator); import Data.Function (on)
a226555 n = numerator $ sum $
            zipWith ((%) `on` toInteger) (a226314_row n) (a054531_row n)

A278635 Least x such that ax-1 and bx-1 are simultaneously prime and n (sequence index) is the position of rational a/b (lowest form) in the triangular array A226314(n)/A054531(n).

Original entry on oeis.org

1, 3, 3, 4, 2, 4, 3, 2, 1, 3, 4, 4, 4, 6, 4, 3, 1, 4, 4, 4, 3, 6, 2, 2, 2, 6, 2, 6, 3, 6, 1, 2, 4, 2, 6, 3, 6, 2, 2, 2, 6, 1, 2, 6, 6, 3, 4, 2, 6, 2, 4, 2, 6, 2, 3, 4, 4, 4, 12, 4, 4, 12, 4, 10, 18, 4, 4, 2, 2, 2, 4, 4, 2, 4, 12, 4, 4, 4, 8, 24, 8, 8, 18, 8, 14, 24, 8, 8, 18

Offset: 1

Frank M Jackson and Michael B Rees, Nov 24 2016

nonn

Rationals a/b (lowest form) can be mapped uniquely into a triangular array A226314(n)/A054531(n).

By Schinzel's Hypothesis H the functions f_1=ax-1 and f_2=bx-1 have infinite values of x such that f_1 and f_2 are simultaneously prime. Hence a/b can be expressed using two primes p and q as a/b=(p+1)/(q+1). This sequence determines the least x for generating p=f_1 and q=f_2 with the sequence index n selecting a/b from the triangular array A226314(n)/A054531(n).

			a(7)=3 because A226314(7)/A054531(7)=1/4 and with x=3 we have p=f_1=x-1=2 and q=f_2=4x-1=11. Therefore (p+1)/(q+1)=3/12=1/4.

Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica IV., 1957.
Wikipedia Schinzel's hypothesis H.

Cf. A226314, A054531.

func[{i_, j_}] := {j(j-1)/2+i->{j+(i-j)/GCD[i, j], j/GCD[i, j]}}; getx[{a_, b_}] := Module[{f1, f2, x}, If[a==b, 1, (f1=a*x-1; f2=b*x-1; x = 1; While[(!PrimeQ[f1]||!PrimeQ[f2])&&x<10^5, x++]; If[x==10^5, Abort[], x])]]; assoc=Association@Flatten[Table[func[{a, b}], {b, 1000}, {a, b}], 1]; Table[getx[assoc[n]], {n, 1, 100}]

A242061 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1, x+y odd and for the least y, {x, y} are integers such that x*y(y^2-x^2)/A006991(n) is a perfect square.

Original entry on oeis.org

14, 2, 129, 52686, 29, 7, 9, 1274, 296125969, 12012350, 5, 1279281, 44, 302583265614, 780914, 90, 316, 2605, 106023820090609, 1754402265205275806, 7794, 72957466300254, 768323201, 40, 18505, 23, 6478321, 3966329, 326, 14280500082452241

Offset: 1

Frank M Jackson, Aug 13 2014

nonn

The triangle array A226314(n)/A054531(n) that enumerates all positive rationals x/y can be generalized to enumerate all ordered pairs {x, y} where x and y are natural numbers. For example, A243808 uses a subset of this triangular array to enumerate all primitive Pythagorean triples (PPT).

A006991(n) is the sequence of primitive congruent numbers and by definition there exists a PPT whose area is equal to k^2*A006991(n) for some integer k. a(n) is an enumeration of these PPT's and is a measure of the number of Pythagorean triangles that have to be searched to find a PPT with the least hypotenuse that has an area equal to k^2*A006991(n). If {x, y} are the generators of a PPT (a, b, c) where a = y^2-x^2, b = 2x*y, c=y^2+x^2 then its area = x*y(y^2-x^2). The Mathematica program limits searches to the first 12.5 million generated PPT's. All other results have been obtained from tables cataloged by Hisanori Mishima (see Links).

			.  j       {A226314(n),A054531(n)}, 1<=i<=j<=12 and n=i+j(j-1)/2
.  --   --------------------------------------------------------
.   1:  1,1
.   2:  1,2 2,1
.   3:  1,3 2,3 3,1
.   4:  1,4 3,2 3,4 4,1
.   5:  1,5 2,5 3,5 4,5 5,1
.   6:  1,6 4,3 5,2 5,3 5,6 6,1
.   7:  1,7 2,7 3,7 4,7 5,7 6,7 7,1
.   8:  1,8 5,4 3,8 7,2 5,8 7,4 7,8 8,1
.   9:  1,9 2,9 7,3 4,9 5,9 8,3 7,9 8,9 9,1
.  10:  1,10 6,5 3,10 7,5 9,2 8,5 7,10 9,5 9,10 10,1
.  11:  1,11 2,11 3,11 4,11 5,11 6,11 7,11 8,11 9,11 10,11 11,1
.  12:  1,12 7,6 9,4 10,3 5,12 11,2 7,12 11,3 11,4 11,6 11,12 12,1 .
a(13)=44 and A006991(13)=34 so 34 is the 13th congruent number. a(13) gives the 44th term of the triangular array at index (8, 9). This gives (x, y) as (8, 9), it generates the PPT (17, 144, 145) and has an area 6^2*34 = 1224.

Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
Hisanori Mishima, 361 Congruent Numbers g: 1<=g<=999, Mathematician's Secret Room, Chapter 10 : Congruent Numbers (D27 Congruent numbers).
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A006991, A054531, A226314.

lst1={5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 34, 37, 38, 39, 41, 46, 47, 53, 55, 61, 62, 65, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95, 101}; getpos[n0_] := (lst=0; Do[If[IntegerQ@Sqrt[m*n(m-n)(m+n)/n0]&&OddQ[m+n] && GCD[m, n]==1, (lst=m(m-1)/2+n; Break[])], {m, 2, 5000}, {n, 1, m-1}]; lst); SetAttributes[getpos, Listable]; getpos[lst1]

A279369 A mapping of rationals a/b (lowest form) to prime rationals p/q such that a/b = (p+1)/(q+1), where n (the sequence index) selects the rationals a/b from the triangle array A226314(n)/A054531(n) and a(n) selects the prime rationals p/q from the same array.

Original entry on oeis.org

1, 12, 18, 58, 13, 74, 57, 19, 5, 72, 174, 178, 182, 429, 217, 138, 8, 225, 247, 272, 162, 825, 81, 83, 85, 849, 89, 999, 255, 1047, 23, 110, 484, 103, 1122, 288, 1383, 139, 114, 143, 1407, 32, 149, 1425, 1518, 408, 711, 176, 1677, 165, 727, 184, 1701, 188, 450, 906, 910, 914

Offset: 1

Frank M Jackson and Michael B Rees, Dec 10 2016

nonn

Rationals a/b (lowest form) can be mapped 1-to-1 to a positive integer n where a/b is the n-th term of the triangular array A226314(n)/A054531(n). Consider two function of x, f_1 = ax-1 and f_2 = bx-1. Then by Schinzel's Hypothesis H there are infinite values of x such that f_1 and f_2 are simultaneously prime allowing a/b to be expressed using two primes p and q as a/b=(p+1)/(q+1).

By choosing the least x for generating p=f_1 and q=f_2 (see A278635) it is possible to find a unique prime rational p/q that maps to rational a/b. If n is the sequence index that selects the rational a/b from the triangular array A226314(n)/A054531(n), then a(n) selects the prime rationals p/q from the same array.

			a(7)=57 because A226314(7)/A054531(7)=1/4 and with least x=3 we have p=f_1=x-1=2 and q=f_2=4x-1=11. Therefore (p+1)/(q+1)=3/12=1/4. Also A226314(57)/A054531(57)=p/q=2/11.

Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica IV., 1957.
Wikipedia Schinzel's hypothesis H.

Cf. A054531, A226314, A278635.

func[{i_, j_}] := {j(j-1)/2+i->{j+(i-j)/GCD[i, j], j/GCD[i, j]}}; rfunc[{i_, j_}] := {{j+(i-j)/GCD[i, j], j/GCD[i, j]}->j(j-1)/2+i}; getx[{a_, b_}] := Module[{f1, f2, x}, If[a==b, {1, 1}, (f1=a*x-1; f2=b*x-1; x=1; While[(!PrimeQ[f1]||!PrimeQ[f2])&&x<10^5, x++]; If[x==10^5, Abort[], {f1, f2}])]]; assoc=Association@Flatten[Table[func[{a, b}], {b, 1000}, {a, b}], 1]; rassoc=Association@Flatten[Table[rfunc[{a, b}], {b, 1000}, {a, b}], 1]; Table[rassoc[getx[assoc[n]]], {n, 1, 100}]

A040001 1 followed by {1, 2} repeated.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane

Continued fraction for sqrt(3).
Also coefficient of the highest power of q in the expansion of the polynomial nu(n) defined by: nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,1), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
nu(0)=1 nu(1)=1; nu(2)=2; nu(3)=3+q; nu(4)=5+3q+2q^2; nu(5)=8+7q+6q^2+4q^3+q^4; nu(6)=13+15q+16q^2+14q^3+11q^4+5q^5+2q^6.
From Jaroslav Krizek, May 28 2010: (Start)
a(n) = denominators of arithmetic means of the first n positive integers for n >= 1.
See A026741(n+1) or A145051(n) - denominators of arithmetic means of the first n positive integers. (End)
From R. J. Mathar, Feb 16 2011: (Start)
This is a prototype of multiplicative sequences defined by a(p^e)=1 for odd primes p, and a(2^e)=c with some constant c, here c=2. They have Dirichlet generating functions (1+(c-1)/2^s)*zeta(s).
Examples are A153284, A176040 (c=3), A040005 (c=4), A021070, A176260 (c=5), A040011, A176355 (c=6), A176415 (c=7), A040019, A021059 (c=8), A040029 (c=10), A040041 (c=12). (End)
a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A000325(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
For n > 0: denominators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n; see A226555 for numerators. - Reinhard Zumkeller, Jun 10 2013
From Jianing Song, Nov 01 2022: (Start)
For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.
The maximal gap is given by A048669. (End)

			1.732050807568877293527446341... = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...))))
G.f. = 1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + ...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Ashok Kumar Gupta and Ashok Kumar Mittal, Bifurcating continued fractions, arXiv:math/0002227 [math.GM] (2000).
Michael Somos, Rational Function Multiplicative Coefficients.
Eric Weisstein's World of Mathematics, Square Root.
Eric Weisstein's World of Mathematics, Theodorus's Constant.
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000034, A002194, A133566, A083329 (binomial Transf).

Apart from a(0) the same as A134451.

a040001 0 = 1; a040001 n = 2 - mod n 2
a040001_list = 1 : cycle [1, 2]  -- Reinhard Zumkeller, Apr 16 2015

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[3],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{1},120,{2,1}] (* Harvey P. Dale, Nov 26 2015 *)

{a(n) = 2 - (n==0) - (n%2)} /* Michael Somos, Jun 11 2003 */

{ allocatemem(932245000); default(realprecision, 12000); x=contfrac(sqrt(3)); for (n=0, 20000, write("b040001.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

Multiplicative with a(p^e) = 2 if p even; 1 if p odd. - David W. Wilson, Aug 01 2001
G.f.: (1 + x + x^2) / (1 - x^2). E.g.f.: (3*exp(x)-2*exp(0)+exp(-x))/2. - Paul Barry, Apr 27 2003
a(n) = (3-2*0^n +(-1)^n)/2. a(-n)=a(n). a(2n+1)=1, a(2n)=2, n nonzero.
a(n) = sum{k=0..n, F(n-k+1)*(-2+(1+(-1)^k)/2+C(2, k)+0^k)}. - Paul Barry, Jun 22 2007
Row sums of triangle A133566. - Gary W. Adamson, Sep 16 2007
Euler transform of length 3 sequence [ 1, 1, -1]. - Michael Somos, Aug 04 2009
Moebius transform is length 2 sequence [ 1, 1]. - Michael Somos, Aug 04 2009
a(n) = sign(n) + ((n+1) mod 2) = 1 + sign(n) - (n mod 2). - Wesley Ivan Hurt, Dec 13 2013

A050873 Triangular array T read by rows: T(n,k) = gcd(n,k).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

The function T(n,k) = T(k,n) is defined for all integer k,n but only the values for 1 <= k <= n as a triangular array are listed here.
For each divisor d of n, the number of d's in row n is phi(n/d). Furthermore, if {a_1, a_2, ..., a_phi(n/d)} is the set of positive integers <= n/d that are relatively prime to n/d then T(n,a_i * d) = d. - Geoffrey Critzer, Feb 22 2015
Starting with any row n and working downwards, consider the infinite rectangular array with k = 1..n. A repeating pattern occurs every A003418(n) rows. For example, n=3: A003418(3) = 6. The 6-row pattern starting with row 3 is {1,1,3}, {1,2,1}, {1,1,1}, {1,2,3}, {1,1,1}, {1,2,1}, and this pattern repeats every 6 rows, i.e., starting with rows {9,15,21,27,...}. - Bob Selcoe and Jamie Morken, Aug 02 2017

			Rows:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 1, 4;
  1, 1, 1, 1, 5;
  1, 2, 3, 2, 1, 6; ...

T. D. Noe, Rows n=1..100, flattened
Marcelo Polezzi, A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445-446.
Eric Weisstein's World of Mathematics, Greatest Common Divisor
Wikipedia, Greatest Common Divisor

Cf. A003989.
Cf. A002262, A054531, A226314.
Cf. A018804 (row sums), A245717.
Cf. A132442 (sums of divisors).
Cf. A003418.

a050873 = gcd
a050873_row n = a050873_tabl !! (n-1)
a050873_tabl = zipWith (map . gcd ) [1..] a002260_tabl
-- Reinhard Zumkeller, Dec 12 2015, Aug 13 2013, Jun 10 2013

ColumnForm[Table[GCD[n, k], {k, 12}, {n, k}], Center] (* Alonso del Arte, Jan 14 2011 *)

{T(n, k) = gcd(n, k)} /* Michael Somos, Jul 18 2011 */

a(n) = gcd(A002260(n), A002024(n)); A054521(n) = A000007(a(n)). - Reinhard Zumkeller, Dec 02 2009
T(n,k) = A075362(n,k)/A051173(n,k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011
T(n, k) = T(k, n) = T(-n, k) = T(n, -k) = T(n, n+k) = T(n+k, k). - Michael Somos, Jul 18 2011
T(n,k) = A051173(n,k) / A051537(n,k). - Reinhard Zumkeller, Jul 07 2013

A054531 Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2000

Sum of n-th row = A057660(n). - Reinhard Zumkeller, Aug 12 2009
Read as a linear sequence, this is conjectured to be the length of the shortest cycle of pebble-moves among the partitions of n (cf. A201144). - Andrew V. Sutherland, Nov 27 2011
The triangle of fractions A226314(i,j)/A054531(i,j) is an efficient way to enumerate the rationals [Fortnow]. - N. J. A. Sloane, Jun 09 2013

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

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
R. J. Mathar, Plots of cycle graphs of the finite groups up to order 36, (2015).
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A050873, A164306, A226314, A277227 (row reversed, k=0..n-1).

a054531 n k = div n $ gcd n k
a054531_row n = a054531_tabl !! (n-1)
a054531_tabl = zipWith (\u vs -> map (div u) vs) [1..] a050873_tabl
-- Reinhard Zumkeller, Jun 10 2013

Table[n/GCD[n,k], {n,1,10}, {k,1,n}]//Flatten (* G. C. Greubel, Sep 13 2017 *)

for(n=1,10, for(k=1,n, print1(n/gcd(n,k), ", "))) \\ G. C. Greubel, Sep 13 2017

cp's OEIS Frontend

A243808 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1 and x+y odd.

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A226555 Numerators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n.

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Haskell

A278635 Least x such that ax-1 and bx-1 are simultaneously prime and n (sequence index) is the position of rational a/b (lowest form) in the triangular array A226314(n)/A054531(n).

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A242061 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1, x+y odd and for the least y, {x, y} are integers such that x*y(y^2-x^2)/A006991(n) is a perfect square.

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A279369 A mapping of rationals a/b (lowest form) to prime rationals p/q such that a/b = (p+1)/(q+1), where n (the sequence index) selects the rationals a/b from the triangle array A226314(n)/A054531(n) and a(n) selects the prime rationals p/q from the same array.

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A040001 1 followed by {1, 2} repeated.

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PARI

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A050873 Triangular array T read by rows: T(n,k) = gcd(n,k).

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A054531 Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).