A221918 -id:A221918 - cp's OEIS frontend

A060294 Decimal expansion of Buffon's constant 2/Pi.

6, 3, 6, 6, 1, 9, 7, 7, 2, 3, 6, 7, 5, 8, 1, 3, 4, 3, 0, 7, 5, 5, 3, 5, 0, 5, 3, 4, 9, 0, 0, 5, 7, 4, 4, 8, 1, 3, 7, 8, 3, 8, 5, 8, 2, 9, 6, 1, 8, 2, 5, 7, 9, 4, 9, 9, 0, 6, 6, 9, 3, 7, 6, 2, 3, 5, 5, 8, 7, 1, 9, 0, 5, 3, 6, 9, 0, 6, 1, 4, 0, 3, 6, 0, 4, 5, 5, 2, 1, 1, 0, 6, 5, 0, 1, 2, 3, 4, 3, 8, 2, 4, 2, 9, 1

Offset: 0

Jason Earls, Mar 28 2001

The probability P(l,d) that a needle of length l will land on a line, given a floor with equally spaced parallel lines at a distance d (>=l) apart, is (2/Pi)*(l/d). - Benoit Cloitre, Oct 14 2002
Lim_{n->infinity} z(n)/log(n) = 2/Pi, where z(n) is the expected number of real zeros of a random polynomial of degree n with real coefficients chosen from a standard Gaussian distribution (cf. Finch reference). - Benoit Cloitre, Nov 02 2003
Also the ratio of the average chord length when two points are chosen at random on a circle of radius r to the maximum possible chord length (i.e., diameter) = A088538*r / (2*r) = 2/Pi. Is there a (direct or obvious) relationship between this fact and that 2/Pi is the "magic geometric constant" for a circle (see MathWorld link)? - Rick L. Shepherd, Jun 22 2006
Blatner (1997) says that Euler found a "fascinating infinite product" for Pi involving the prime numbers, but the number he then describes does not match Pi. Switching the numerator and the denominator results in this number. - Alonso del Arte, May 16 2012
2/Pi is also the height (the ordinate y) of the geometric centroid of each arbelos (see the references and links given under A221918) with a large radius r=1 and any small ones r1 and r2 = 1 - r1, for 0 < r1 < 1. Use the integral formula given, e.g., in the MathWorld or Wikipedia centroid reference, for the two parts of the arbelos (dissected by the vertical line x = 2*r1), and then use the decomposition formula. The heights y1 and y2 of the centroids of the two parts satisfy: F1(r1)*y1(r1) = 2*r1^2*(1-r1) and F2(1-r1)*y2(1-r1) = 2*(1-r1)^2*r1. The r1 dependent area F = F1 + F2 is Pi*r1*(1-r1). (F1 and F2 are rather complicated but their explicit formulas are not needed here.) The r1 dependent horizontal coordinate x with origin at the left tip of the arbelos is x = r1 + 1/2. - Wolfdieter Lang, Feb 28 2013
Construct a quadrilateral of maximal area inside a circle. The quadrilateral is necessarily an inscribed square (with diagonals that are diameters). 2/Pi is the ratio of the square's area to the circle's area. - Rick L. Shepherd, Aug 02 2014
The expected number of real roots of a real polynomial of degree n varies as this constant times the (natural) logarithm of n, see Kac, when its coefficients are chosen from the standard uniform distribution. This may be related to Rick Shepherd's comment. - Charles R Greathouse IV, Oct 06 2014
2/Pi is also the minimum value, at x = 1/2, on (0,1) of 1/(Pi*sqrt(x*(1-x))), the nonzero piece of the probability density function for the standard arcsine distribution. - Rick L. Shepherd, Dec 05 2016
The average distance from the center of a unit-radius circle to the midpoints of chords drawn between two points that are uniformly and independently chosen at random on the circumference of the circle. - Amiram Eldar, Sep 08 2020
2/Pi <= sin(x)/x < 1 for 0 < |x| <= Pi/2 is Jordan's inequality, also known as (2/Pi) * x <= sin(x) <= x for 0 <= x <= Pi/2; this inequality was named after the French mathematician Camille Jordan (1838-1922). - Bernard Schott, Jan 07 2023
This constant 2/Pi was named after the needle experiment, described in 1777 by the French naturalist and mathematician Georges-Louis Leclerc, Comte de Buffon (1707-1788). Note that the parrot Buffon's macaw and the antelope Buffon's kob were named also after Buffon. - Bernard Schott, Jan 10 2023
2*n*log(n)/Pi is also the dominant term in the asymptotic expansion of Sum_{k=1..n-1} csc(Pi*k/n) at n tending to infinity. - Iaroslav V. Blagouchine, Apr 21 2025

			2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...

David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.
G. Buffon, Essai d'arithmétique morale. Supplément à l'Histoire Naturelle, Vol. 4, 1777.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 141, 539.
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, p. 7, eq. (1.2) and p. 105 eq. (7.4.2) with s=1/2.
Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, 1991.
Daniel A. Klain and Gian-Carlo Rota, Introduction to Geometric Probability, Cambridge, 1997, see Chap. 1.
Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 53.
Robert M. Young, Excursions in Calculus, An Interplay of the Continuous and the Discrete. Dolciani Mathematical Expositions Number 13. MAA.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Iaroslav V. Blagouchine and Eric Moreau, On a finite sum of cosecants appearing in various problems, Journal of Mathematical Analysis and Applications, vol. 539, no. 1, pt. 2, pp. 1-36, 2024. See p. 18.
K. S. Brown, MathPages: The Algebra of an Infinite Grid of Resistors
G. Buffon, Essai d'arithmétique morale, Supplément à l'Histoire Naturelle, Vol. 4, 1777.
Encyclopedia of Mathematics, Arcsine distribution
Boris Gourevitch, L'univers de Pi
Mark Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:4 (1943), pp. 314-320.
MacTutor History of Mathematics archive, Georges-Louis Leclerc, Comte de Buffon.
Veikko Nevanlinna, On constants connected with the prime number theorem for arithmetic progressions, Annales Academiae Scientiarum Fennicae Ser. A. I., No. 539 (1973).
Da-Wei Niu, Jian Cao, and Feng Qi, Generalizations of Jordan's inequality and concerned relations, U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2010
Herbert Solomon, Geometric Probability, SIAM, 1978, p. 152. [See average chord length comment]
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Magic Geometric Constants.
Eric Weisstein's World of Mathematics, Prime Products.
Eric Weisstein's World of Mathematics, Geometric Centroid.
Eric Weisstein's World of Mathematics, Jordan's Inequality.
Wikipedia, Buffon's needle problem.
Wikipedia, Centroid.
Wikipedia, Jordan's inequality.
Index entries for transcendental numbers.

Cf. A000796 (Pi), A088538, A154956, A082542 (numerators in an infinite product), A053300 (continued fraction without the initial 0).

Cf. A076668 (sqrt(2/Pi)).

R:= RealField(100); 2/Pi(R); // G. C. Greubel, Mar 09 2018

Digits:=100: evalf(2/Pi); # Wesley Ivan Hurt, Aug 02 2014

Mathematica
```
RealDigits[ N[ 2/Pi, 111]][[1]]
```

default(realprecision, 20080); x=20/Pi; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b060294.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009

2/Pi = 1 - 5*(1/2)^3 + 9*((1*3)/(2*4))^3 - 13*((1*3*5)/(2*4*6))^3 ... - Jason Earls [formula corrected by Paul D. Hanna, Mar 23 2013]
The preceding formula is 2/Pi = Sum_{n>=0} (-1)^n * (4*n+1) * Product_{k=1..n} (2*k-1)^3/(2*k)^3. - Alexander R. Povolotsky, Mar 24 2013. [See the Hardy reference. - Wolfdieter Lang, Nov 13 2016]
2/Pi = Product_{n>=2} (p(n) + 2 - (p(n) mod 4))/p(n), where p(n) is the n-th prime. - Alonso del Arte, May 16 2012
2/Pi = Sum_{k>=0} ((2*k)!/(k!)^2)^3*((42*k+5)/(2^{12*k+3})) (due to Ramanujan). - L. Edson Jeffery, Mar 23 2013
Equals sinc(Pi/2). - Peter Luschny, Oct 04 2019
From A.H.M. Smeets, Apr 11 2020: (Start)
Equals Product_{i > 0} cos(Pi/2^(i+1)).
Equals Product_{i > 0} f_i(2)/2, where f_0(2) = 0, f_(i+1)(2) = sqrt(2+f_i(2)) for i >= 0; a formula by François Viète (16th century).
Note that cos(Pi/2^(i+1)) = f_i(2)/2, i >= 0. (End)
Equals lim_{n->infinity} (1/n) * Sum_{k=1..n} abs(sin(k * m)) for all nonzero integers m (conjectured). Works with cos also. - Dimitri Papadopoulos, Jul 17 2020
From Amiram Eldar, Sep 08 2020: (Start)
Equals Product_{k>=1} (1 - 1/(2*k)^2).
Equals lim_{k->oo} (2*k+1)*binomial(2*k,k)^2/2^(4*k).
Equals Sum_{k>=0} binomial(2*k,k)^2/((2*k+2)*2^(4*k)). (End)
Equals Sum_{k>=0} mu(4*k+1)/(4*k+1) (Nevanlinna, 1973). - Amiram Eldar, Dec 21 2020
Equals 1 - Sum_{n >= 1} (1/16^n) * binomial(2*n, n)^2 * 1/(2*n - 1). See Young, p. 264. - Peter Bala, Feb 17 2024
Equals binomial(0, 1/2) = binomial(0, -1/2). - Peter Luschny, Dec 05 2024
From Peter Bala, Dec 10 2024:(Start)
2/Pi =  1 - 1/(2 + 2/(1 + 6/(1 + 12/(1 + 20/(1 + ... + n*(n+1)/(1 + ...), a continued fraction representation due to Euler. See A346943.
Equals 1 - (1/2)*Sum_{n >= 0} A005566(n)*(-1/4)^n. (End)

A227041 Triangle of numerators of harmonic mean of n and m, 1 <= m <= n.

Original entry on oeis.org

1, 4, 2, 3, 12, 3, 8, 8, 24, 4, 5, 20, 15, 40, 5, 12, 3, 4, 24, 60, 6, 7, 28, 21, 56, 35, 84, 7, 16, 16, 48, 16, 80, 48, 112, 8, 9, 36, 9, 72, 45, 36, 63, 144, 9, 20, 10, 60, 40, 20, 15, 140, 80, 180, 10, 11, 44, 33, 88, 55, 132, 77, 176, 99, 220, 11

Offset: 1

Wolfdieter Lang, Jul 01 2013

The harmonic mean H(n,m) is the reciprocal of the arithmetic mean of the reciprocals of n and m: H(n,m) = 1/((1/2)*(1/n +1/m)) = 2*n*m/(n+m). 1/H(n,m) marks the middle of the interval [1/n, 1/m] if m < n: 1/H(n,m) = 1/n + (1/2)*(1/m - 1/n). For m < n one has m < H(n,m) < n, and H(n,n) = n.
  H(n,m) = H(m,n).
For the rationals H(n,m)/2 see A221918(n,m)/A221919(n,m). See the comments under A221918.

			The triangle of numerators of H(n,m), called a(n,m) begins:
n\m  1   2   3   4   5    6    7    8    9   10  11 ...
1:   1
2:   4   2
3:   3  12   3
4:   8   8  24   4
5:   5  20  15  40   5
6:  12   3   4  24  60    6
7:   7  28  21  56  35   84    7
8:  16  16  48  16  80   48  112    8
9:   9  36   9  72  45   36   63  144    9
10: 20  10  60  40  20   15  140   80  180   10
11: 11  44  33  88  55  132   77  176   99  220  11
...
a(4,3) = numerator(24/7) = 24 = 24/gcd(7,18).
The triangle of the rationals H(n,m) begins:
n\m    1      2     3     4      5      6      7      8   9
1:   1/1
2:   4/3    2/1
3:   3/2   12/5   3/1
4:   8/5    8/3  24/7   4/1
5:   5/3   20/7  15/4  40/9    5/1
6:  12/7    3/1   4/1  24/5  60/11    6/1
7:   7/4   28/9  21/5 56/11   35/6  84/13    7/1
8:  16/9   16/5 48/11  16/3  80/13   48/7 112/15    8/1
9:   9/5  36/11   9/2 72/13   45/7   36/5   63/8 144/17 9/1
...
H(4,3) = 2*4*3/(4 + 3) = 2*4*3/7 = 24/7.

Eric Weisstein's World of Mathematics, Harmonic Mean.

Cf. A227042, A022998 (m=1), A227043 (m=2), A227106 (m=3), A227107 (m=4), A221918/A221919.

a(n,m) = numerator(2*n*m/(n+m)), 1 <= m <= n.

a(n,m) = 2*n*m/gcd(n+m,2*n*m) = 2*n*m/gcd(n+m,2*m^2), n >= 0.

A227042 Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.

Original entry on oeis.org

1, 3, 1, 2, 5, 1, 5, 3, 7, 1, 3, 7, 4, 9, 1, 7, 1, 1, 5, 11, 1, 4, 9, 5, 11, 6, 13, 1, 9, 5, 11, 3, 13, 7, 15, 1, 5, 11, 2, 13, 7, 5, 8, 17, 1, 11, 3, 13, 7, 3, 2, 17, 9, 19, 1, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 1

Offset: 1

Wolfdieter Lang, Jul 01 2013

See the comments under A227041. a(n,m) gives the denominator of H(n,m) = 2*n*m/(n+m) in lowest terms.

			The triangle of denominators of H(n,m), called a(n,m) begins:
n\m  1   2   3   4   5    6    7    8    9   10  11 ...
1:   1
2:   3   1
3:   2   5   1
4:   5   3   7   1
5:   3   7   4   9   1
6:   7   1   1   5  11    1
7:   4   9   5  11   6   13    1
8;   9   5  11   3  13    7   15    1
9:   5  11   2  13   7    5    8   17    1
10: 11   3  13   7   3    2   17    9   19    1
11:  6  13   7  15   8   17    9   19   10   21   1
...
For the triangle of the rationals H(n,m) see the example section of A227041.
H(4,2) = denominator(16/6) = denominator(8/3) = 3 = 6/gcd(6,8) = 6/2.

Eric Weisstein's World of Mathematics, Harmonic Mean.

Cf. A227041, A026741 (column m=1), A000265 (m=2), A106619 (m=3), A227140(n+8) (m=4), A227108 (m=5), A221918/A221919.

a(n,m) = denominator(2*n*m/(n+m)), 1 <= m <= n.

a(n,m) = (n+m)/gcd(2*n*m, n+m) = (n+m)/gcd(n+m, 2*m^2), 1 <= m <= n.

A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

Original entry on oeis.org

2, 3, 1, 4, 5, 2, 5, 3, 7, 1, 6, 7, 8, 9, 2, 7, 2, 1, 5, 11, 1, 8, 9, 10, 11, 12, 13, 2, 9, 5, 11, 3, 13, 7, 15, 1, 10, 11, 4, 13, 14, 5, 16, 17, 2, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1

Offset: 1

Wolfdieter Lang, Feb 21 2013

The triangle of the corresponding denominators is given in A221918.
See A221918 for comments on resistance, reduced mass and radius of the twin circles in Archimedes's arbelos, as well as references.
The column sequences give A000027(n+1), A060819(n+2), A106610(n+3), A106617(n+4), A132739(n+5), A222464 for n >= m = 1,2,..., 6.

			The triangle a(n,m) begins:
n\m   1   2    3   4    5   6    7    8   9   10  11  12 ...
1:    2
2:    3   1
3:    4   5    2
4:    5   3    7   1
5:    6   7    8   9    2
6:    7   2    1   5   11   1
7:    8   9   10  11   12  13    2
8:    9   5   11   3   13   7   15    1
9:   10  11    4  13   14   5   16   17
10:  11   3   13   7    3   4   17    9  19    1
11:  12  13   14  15   16  17   18   15  20   21   2
12:  13   7    5   1   17   1   19    5   7   11  23   1
...
a(n,1) = n + 1 because R(n,1) = n/(n+1), gcd(n,n+1) = 1, hence denominator(R(n,m)) = n + 1.
a(5,4) = 9 because R(5,4) = 20/9, gcd(20,9) = 1, hence denominator( R(5,4)) = 9.
a(6,3) = 1 because R(6,3) = 18/9 = 2/1.
For the rationals R(n,m) see A221918.

Cf. A221918 (companion triangle).

a[n_, m_] := Numerator[1/n + 1/m]; Table[a[n, m], {n, 1, 12}, {m, 1, n}] // Flatten  (* Jean-François Alcover, Feb 25 2013 *)

a(n,m) = numerator(2/n + 1/m), n >= m >= 1, and 0 otherwise.

A221918(n,m)/a(n,m) = R(n,m) = n*m/(n+m). 1/R(n,m) = 1/n + 1/m.

A221920 a(n) = 3n/gcd(3n, n+3), n >= 3.

Original entry on oeis.org

3, 12, 15, 2, 21, 24, 9, 30, 33, 12, 39, 42, 5, 48, 51, 18, 57, 60, 21, 66, 69, 8, 75, 78, 27, 84, 87, 30, 93, 96, 11, 102, 105, 36, 111, 114, 39, 120, 123, 14, 129, 132, 45, 138, 141, 48, 147, 150, 17, 156, 159, 54, 165, 168, 57, 174, 177, 20, 183, 186, 63

Offset: 3

Wolfdieter Lang, Feb 21 2013

This is the third column sequence (m = 3) of the triangle A221918.

			a(6) = numerator(18/9) = numerator(2) = 2 = 18/gcd(18,9) = 18/9 = 18/gcd(9,9) = 18/9.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

Cf. A221918, A000027 (m=1), A145979(m=2), A221921 (m=4), A222463 (m=5).

a[n_] := 3*n/GCD[3*n, n+3]; Array[a, 63, 3] (* Amiram Eldar, Oct 09 2023 *)

a(n)=3*n/gcd(3*n,n+3) \\ Charles R Greathouse IV, Apr 18 2013

a(n) = A221918(n,3) = numerator(n*3/(n+3)), n >= 3.
a(n) = 3*n/gcd(3*n,n+3), n >= 3.
a(n) = 3*n/gcd(9,n+3), n >= 3, (because gcd(n+3,3*n) = gcd(n+3,3*n - 3*(n+3)) = gcd(n+3,-3^2) = gcd(n+3,9)).
G.f.: -x^3*(6*x^17 + 3*x^16 - 3*x^14 - 6*x^13 - x^12 - 12*x^11 - 15*x^10 - 6*x^9 - 33*x^8 - 30*x^7 - 9*x^6 - 24*x^5 - 21*x^4 - 2*x^3 - 15*x^2 - 12*x - 3) / ((x-1)^2*(x^2 + x + 1)^2*(x^6 + x^3 + 1)^2). - Colin Barker, Feb 25 2013
Sum_{k=3..n} a(k) ~ (61/54) * n^2. - Amiram Eldar, Oct 09 2023

A221921 a(n) = 4n/gcd(4n,n+4), n >= 4.

Original entry on oeis.org

2, 20, 12, 28, 8, 36, 20, 44, 3, 52, 28, 60, 16, 68, 36, 76, 10, 84, 44, 92, 24, 100, 52, 108, 7, 116, 60, 124, 32, 132, 68, 140, 18, 148, 76, 156, 40, 164, 84, 172, 11, 180, 92, 188, 48, 196, 100, 204, 26, 212, 108, 220, 56, 228, 116, 236, 15, 244, 124, 252, 64

Offset: 4

Wolfdieter Lang, Feb 21 2013

This is the fourth column of the triangle A221918.

			a(8) = numerator(32/12) = numerator(8/3) = 8 = 32/gcd(32,12) = 32/4 = 32/gcd(16,12).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A221918, A000027 (m=1), A145979(m=2), A221920 (m=3).

Table[(4n)/GCD[4n,n+4],{n,4,70}] (* Harvey P. Dale, May 15 2018 *)

a(n)=4*n/gcd(4*n,n+4) \\ Charles R Greathouse IV, Apr 18 2013

a(n) = A221918(n,4) = numerator(n*4/(n+4)), n >= 4.
a(n) = 4*n/gcd(16,n+4), n >= 4.
G.f.: x^4*(-12*x^31 - 4*x^30 - 4*x^29 + 4*x^27 + 4*x^26 + 12*x^25 + x^24 + 20*x^23 + 12*x^22 + 28*x^21 + 8*x^20 + 36*x^19 + 20*x^18 + 44*x^17 + 6*x^16 + 76*x^15 + 36*x^14 + 68*x^13 + 16*x^12 + 60*x^11 + 28*x^10 + 52*x^9 + 3*x^8 + 44*x^7 + 20*x^6 + 36*x^5 + 8*x^4 + 28*x^3 + 12*x^2 + 20*x + 2) / (x^32 - 2*x^16 + 1). - Colin Barker, Feb 25 2013
Sum_{k=4..n} a(k) ~ (171/128) * n^2. - Amiram Eldar, Oct 09 2023

A222463 a(n) = n5/gcd(n5,n+5), n >= 5.

Original entry on oeis.org

5, 30, 35, 40, 45, 10, 55, 60, 65, 70, 15, 80, 85, 90, 95, 4, 105, 110, 115, 120, 25, 130, 135, 140, 145, 30, 155, 160, 165, 170, 35, 180, 185, 190, 195, 40, 205, 210, 215, 220, 9, 230, 235, 240, 245, 50, 255, 260, 265, 270, 55, 280, 285, 290, 295, 60

Offset: 5

Wolfdieter Lang, Feb 21 2013

This is the fifth column (m=5) of the triangle A221918.

			a(10) = numerator(50/15) = numerator(10/3) = 10 = 50/gcd(50,15)= 50/5 = 50/gcd(25,15).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A221918, A000027 (m=1), A145979(m=2), A221920 (m=3), A221921 (m=4).

Table[(5n)/GCD[5n,n +5],{n,5,60}] (* Harvey P. Dale, Nov 06 2020 *)

a(n) = A221918(n,5) = numerator(n*5/(n+5)) = n*5/gcd(n*5,n+5) = n*5/gcd(25,n+5), n >= 5.
a(n) = 2*a(n-25)-a(n-50). - Colin Barker, Feb 25 2013
Sum_{k=5..n} a(k) ~ (521/250) * n^2. - Amiram Eldar, Oct 09 2023

A338797 Triangle read by rows: T(n,k) is the least m such that there exist positive integers x, y and z satisfying x/n + y/k = z/m where all fractions are reduced; 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 4, 12, 1, 5, 10, 15, 20, 1, 6, 3, 2, 12, 30, 1, 7, 14, 21, 28, 35, 42, 1, 8, 8, 24, 8, 40, 24, 56, 1, 9, 18, 9, 36, 45, 18, 63, 72, 1, 10, 5, 30, 20, 2, 15, 70, 40, 90, 1, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1

Offset: 1

Peter Kagey, Nov 09 2020

			Table begins:
  n\k|  1   2   3   4   5   6   7   8   9  10   11 12
  ---+-----------------------------------------------
   1 |  1,
   2 |  2,  1,
   3 |  3,  6,  1,
   4 |  4,  4, 12,  1,
   5 |  5, 10, 15, 20,  1,
   6 |  6,  3,  2, 12, 30,  1,
   7 |  7, 14, 21, 28, 35, 42,  1,
   8 |  8,  8, 24,  8, 40, 24, 56,  1,
   9 |  9, 18,  9, 36, 45, 18, 63, 72,  1,
  10 | 10,  5, 30, 20,  2, 15, 70, 40, 90,   1,
  11 | 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,  1,
  12 | 12, 12,  4,  3, 60,  4, 84, 24, 36, 60, 132, 1.
T(20,10) = 4 because 1/20 + 7/10 = 3/4, and there is no choice of numerators on the left that results in a smaller denominator on the right.

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A051173, A051537, A221918.

import Data.Ratio ((%), denominator)
farey n = [k % n | k <- [1..n], gcd n k == 1]
a338797T n k = minimum [denominator $ a + b | a <- farey n, b <- farey k]

A051537(n,k) <= T(n,k) <= A221918(n,k) <= lcm(n,k) = A051173(n,k).

T(n,k) = lcm(n,k) when gcd(n,k) = 1.

cp's OEIS Frontend

A060294 Decimal expansion of Buffon's constant 2/Pi.

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A227041 Triangle of numerators of harmonic mean of n and m, 1 <= m <= n.

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A227042 Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.

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A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

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A221920 a(n) = 3*n/gcd(3*n, n+3), n >= 3.

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A221921 a(n) = 4*n/gcd(4*n,n+4), n >= 4.

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A222463 a(n) = n*5/gcd(n*5,n+5), n >= 5.

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A338797 Triangle read by rows: T(n,k) is the least m such that there exist positive integers x, y and z satisfying x/n + y/k = z/m where all fractions are reduced; 1 <= k <= n.

A221920 a(n) = 3n/gcd(3n, n+3), n >= 3.

A221921 a(n) = 4n/gcd(4n,n+4), n >= 4.

A222463 a(n) = n5/gcd(n5,n+5), n >= 5.