cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A157865 a(n) is the number of numbers of the form 4n+2 in A082542.

Original entry on oeis.org

2, 2, 1, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1
Offset: 0

Views

Author

Paul Curtz, Mar 08 2009

Keywords

Examples

			There are 2 twos, 2 sixes, 1 ten, 1 fourteen etc. a(6)=0 because the 26 = 4*6+2 is absent in A082542.

Crossrefs

Cf. A082542.

Extensions

Edited and corrected by R. J. Mathar, Mar 15 2009

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

Original entry on oeis.org

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

Views

Author

Jason Earls, Mar 28 2001

Keywords

Comments

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

Examples

			2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...

References

  • 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.

Links

Crossrefs

Cf. A000796 (Pi), A088538, A154956, A082542 (numerators in an infinite product), A053300 (continued fraction without the initial 0).
Cf. A076668 (sqrt(2/Pi)).

Programs

  • Magma
    R:= RealField(100); 2/Pi(R); // G. C. Greubel, Mar 09 2018
  • Maple
    Digits:=100: evalf(2/Pi); # Wesley Ivan Hurt, Aug 02 2014
  • Mathematica
    RealDigits[ N[ 2/Pi, 111]][[1]]
  • 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

Formula

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)

A076342 a(n) = A076340(A000040(n)), real part of primes mapped as defined in A076340, A076341.

Original entry on oeis.org

2, 4, 4, 8, 12, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 60, 60, 68, 72, 72, 80, 84, 88, 96, 100, 104, 108, 108, 112, 128, 132, 136, 140, 148, 152, 156, 164, 168, 172, 180, 180, 192, 192, 196, 200, 212, 224, 228, 228, 232, 240, 240, 252, 256, 264, 268, 272
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Comments

By definition of the map defined in A076340, A076341: 2->(2,0) and p->((floor(p/4)+floor((p mod 4)/2))*4,2-(p mod 4)) for odd primes p.
Number of solutions to x^2 + y^2 = 1 (mod p). - Lekraj Beedassy, Oct 22 2004

Examples

			A000040(11)=31=(32-1) -> (32,-1), therefore a(11)=32 and A070750(11)=-1.

Links

Crossrefs

Cf. A000040, A070750, A076340, A076341, A082542, A088538.

Programs

  • Maple
    f:= proc(n) local p;
  p:= ithprime(n);
  if p mod 4 = 1 then p-1 elif p mod 4 = 3 then p+1 else 2 fi
end proc:
map(f, [$1..100]); # Robert Israel, Dec 26 2016
  • Mathematica
    a[1] = 2; a[n_] := With[{p = Prime[n]}, p - JacobiSymbol[-1, p]]; Array[a, 60] (* Jean-François Alcover, Feb 01 2018, after Lekraj Beedassy *)
a[n_] := Prime[n] - 2 + Mod[Prime[n], 4]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

Formula

a(n) = p-(-1/p) = p+(-1)^{(p+1)/2} for an odd prime p. {(a/b) stands for the value of the Legendre symbol}. - Lekraj Beedassy, Oct 22 2004
From Amiram Eldar, Dec 24 2022: (Start)
a(n) = A000040(n) - A070750(n).
a(n) = A100484(n) - A082542(n).
Product_{n>=1} a(n)/prime(n) = 4/Pi (A088538). (End)
Showing 1-3 of 3 results.

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