Search: a221918 -id:a221918
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A060294
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Decimal expansion of Buffon's constant 2/Pi.
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+10
56
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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, p. 141
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.
G. H. Hardy, Ramanujan, 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.
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LINKS
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FORMULA
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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 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
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
2/Pi = 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
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EXAMPLE
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2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...
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MAPLE
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MATHEMATICA
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RealDigits[ N[ 2/Pi, 111]][[1]]
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PROG
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(PARI) 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
(Magma) R:= RealField(100); 2/Pi(R); // G. C. Greubel, Mar 09 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A227041
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Triangle of numerators of harmonic mean of n and m, 1 <= m <= n.
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+10
7
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A227042
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Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.
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+10
7
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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
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OFFSET
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1,2
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COMMENTS
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See the comments under A227041. a(n,m) gives the denominator of H(n,m) = 2*n*m/(n+m) in lowest terms.
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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STATUS
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approved
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A221919
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Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.
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+10
4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A221920
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a(n) = 3*n/gcd(3*n, n+3), n >= 3.
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+10
3
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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
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OFFSET
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3,1
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COMMENTS
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This is the third column sequence (m = 3) of the triangle A221918.
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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a(6) = numerator(18/9) = numerator(2) = 2 = 18/gcd(18,9) = 18/9 = 18/gcd(9,9) = 18/9.
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MATHEMATICA
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a[n_] := 3*n/GCD[3*n, n+3]; Array[a, 63, 3] (* Amiram Eldar, Oct 09 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A221921
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a(n) = 4*n/gcd(4*n,n+4), n >= 4.
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+10
3
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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
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OFFSET
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4,1
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COMMENTS
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This is the fourth column of the triangle A221918.
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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a(8) = numerator(32/12) = numerator(8/3) = 8 = 32/gcd(32,12) = 32/4 = 32/gcd(16,12).
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MATHEMATICA
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Table[(4n)/GCD[4n, n+4], {n, 4, 70}] (* Harvey P. Dale, May 15 2018 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A222463
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a(n) = n*5/gcd(n*5,n+5), n >= 5.
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+10
2
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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
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OFFSET
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5,1
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COMMENTS
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This is the fifth column (m=5) of the triangle A221918.
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LINKS
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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).
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FORMULA
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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.
Sum_{k=5..n} a(k) ~ (521/250) * n^2. - Amiram Eldar, Oct 09 2023
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EXAMPLE
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a(10) = numerator(50/15) = numerator(10/3) = 10 = 50/gcd(50,15)= 50/5 = 50/gcd(25,15).
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MATHEMATICA
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Table[(5n)/GCD[5n, n +5], {n, 5, 60}] (* Harvey P. Dale, Nov 06 2020 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A338797
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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.
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+10
1
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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T(n,k) = lcm(n,k) when gcd(n,k) = 1.
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EXAMPLE
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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.
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PROG
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(Haskell)
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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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