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 A000796 Decimal expansion of Pi (or digits of Pi). (Formerly M2218 N0880) 930
 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 2, 1, 1, 7, 0, 6, 7, 9, 8, 2, 1, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Sometimes called Archimedes's constant. Ratio of a circle's circumference to its diameter. Also area of a circle with radius 1. Also surface area of a sphere with diameter 1. A useful mnemonic for remembering the first few terms: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ... Also ratio of surface area of sphere to one of the faces of the circumscribed cube. Also ratio of volume of a sphere to one of the six inscribed pyramids in the circumscribed cube. - Omar E. Pol, Aug 09 2012 Also surface area of a quarter of a sphere of radius 1. - Omar E. Pol, Oct 03 2013 Also the area under the peak-shaped even function f(x)=1/cosh(x). Proof: for the upper half of the integral, write f(x) = (2*exp(-x))/(1+exp(-2x)) = 2*Sum_{k>=0} (-1)^k*exp(-(2k+1)*x) and integrate term by term from zero to infinity. The result is twice the Gregory series for Pi/4. - Stanislav Sykora, Oct 31 2013 A curiosity: a 144 X 144 magic square of 7th powers was recently constructed by Toshihiro Shirakawa. The magic sum = 3141592653589793238462643383279502884197169399375105, which is the concatenation of the first 52 digits of Pi. See the MultiMagic Squares link for details. - Christian Boyer, Dec 13 2013 [Comment revised by N. J. A. Sloane, Aug 27 2014] x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 25 2013 Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Jan 13 2014 From Daniel Forgues, Mar 20 2015: (Start) An interesting anecdote about the base-10 representation of Pi, with 3 (integer part) as first (index 1) digit:   358 0   359 3   360 6   361 0   362 0 And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)... (End) Sometimes referred to as Archimedes's constant, because the Greek mathematician computed lower and upper bounds of Pi by drawing regular polygons inside and outside a circle. In Germany it was called the Ludolphian number until the early 20th century after the Dutch mathematician Ludolph van Ceulen (1540-1610) who calculated up to 35 digits of Pi in the late 16th century. - Martin Renner, Sep 07 2016 As of the beginning of 2019 more than 22 trillion decimal digits of Pi are known. See the Wikipedia article "Chronology of computation of Pi". - Harvey P. Dale, Jan 23 2019 On March 14, 2019, Emma Haruka Iwao announced the calculation of 31.4 trillion digits of Pi using Google Cloud's infrastructure. - David Radcliffe, Apr 10 2019 Also volume of three quarters of a sphere of radius 1. - Omar E. Pol, Aug 16 2019 On August 5, 2021, researchers from the University of Applied Sciences of the Grisons in Switzerland announced they had calculated 62.8 trillion digits. Guinness World Records has not verified this yet. - Alonso del Arte, Aug 23 2021 REFERENCES Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998. J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001. P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977. J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997. P. Eyard and J.-P. Lafon, The Number Pi, Amer. Math. Soc., 2004. S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4. Le Petit Archimede, Special Issue On Pi, Supplement to No. 64-5, May 1980 ADCS Amiens. Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 31. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Harry J. Smith, Table of n, a(n) for n = 1..20000 Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, Brendan M. Quine, Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi, arXiv:2004.11711 [math.GM], 2020. Dave Andersen, Pi-Search Page Anonymous, A million digits of Pi Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002. Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005. Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85. Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009, pp. 903-914. D. H. Bailey, On Kanada's computation of 1.24 trillion digits of Pi [archived page] D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, Volume 52, Number 5,  May 2005, pp. 502-514. Harry Baker, "Pi calculated to a record-breaking 62.8 trillion digits", Live Science, August 17, 2021. Steve Baker and Thomas Moore, 100 trillion digits of pi Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379. J. M. Borwein, Talking about Pi J. M. Borwein and M. Macklem, The (Digital) Life of Pi, The Australian Mathematical Society Gazette, Volume 33, Number 5, Sept. 2006, pp. 243-248. Peter Borwein, The amazing number Pi, Nieuw Archief voor de Wiskunde, September 2000, pp. 254-258. Christian Boyer, MultiMagic Squares J. Britton, Mnemonics For The Number Pi [archived page] D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. Jonas Castillo Toloza, Fascinating Method for Finding Pi E. S. Croot, Pade Approximations and the Transcendence of pi L. Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008. L. Euler, De summis serierum reciprocarum, E41. Eureka, Tout pi or not tout pi Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants Jeremy Gibbons, Unbounded Spigot Algorithms for the Digits of Pi X. Gourdon, Pi to 16000 decimals [archived page] Xavier Gourdon, A new algorithm for computing Pi in base 10 X. Gourdon and P. Sebah, Archimedes' constant Pi B. Gourevitch, L'univers de Pi L. Grebelius, Approximation of Pi: First 1000000 digits J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270. Preprint: arXiv:math/0506319 [math.NT] (2005-2006). Carl-Johan Haster, Pi from the sky -- A null test of general relativity from a population of gravitational wave observations, arXiv:2005.05472 [gr-qc], 2020. H. Havermann, Simple Continued Fraction for Pi [archived page] M. D. Huberty et al., 100000 Digits of Pi ICON Project, Pi to 50000 places [archived page] P. Johns, 120000 Digits of Pi [archived page] Yasumasa Kanada, 1.24 trillion digits of Pi Yasumasa Kanada and Daisuke Takahashi, 206 billion digits of Pi [archived page] Literate Programs, Pi with Machin's formula (Haskell) [archived page] Johannes W. Meijer, Pi everywhere poster, Mar 14 2013 J. Moyer, First 10000 digits of pi NERSC, Search Pi [broken link] Remco Niemeijer, The Digits of Pi, programmingpraxis Steve Pagliarulo, Stu's pi page [archived page] Chittaranjan Pardeshi, BBP-Like formula for Pi in Golden Ratio Base Phi Michael Penn, A nice inverse tangent integral., YouTube video, 2020. Michael Penn, Pi is irrational (π∉ℚ), YouTube video, 2020. I. Peterson, A Passion for Pi G. M. Phillips, Table of contents of "Pi: A source Book" Simon Plouffe, 10000 digits of Pi Simon Plouffe, A formula for the nth decimal digit or binary of Pi and powers of Pi, arXiv:2201.12601 [math.NT], 2022. D. Pothet, Chronologie du calcul des decimales de pi [broken link] M. Z. Rafat and D. Dobie, Throwing Pi at a wall, arXiv:1901.06260 [physics.class-ph], 2020. S. Ramanujan, Modular equations and approximations to \pi, Quart. J. Math. 45 (1914), 350-372. M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range, 2013. Grant Sanderson, Why do colliding blocks compute pi?, 3Blue1Brown video (2019). Daniel B. Sedory, The Pi Pages D. Shanks and J. W. Wrench, Jr., Calculation of pi to 100,000 decimals, Math. Comp. 16 1962 76-99. Jean-Louis Sigrist, Les 128000 premieres decimales du nombre PI Sizes, pi A. Sofo, Pi and some other constants, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005. Jonathan Sondow, A faster product for Pi and a new integral for ln Pi/2, arXiv:math/0401406 [math.NT], 2004; Amer. Math. Monthly 112 (2005) 729-734. D. Surendran, Can I have a small container of coffee? [archived page] Wislawa Szymborska, Pi (The admirable number Pi), Miracle Fair, 2002. G. Vacca, A new analytical expression for the number pi, and some historical considerations, Bull. Amer. Math. Soc. 16 (1910), 368-369. Stan Wagon, Is Pi Normal? Eric Weisstein's World of Mathematics, Pi and Pi Digits Wikipedia, Bailey-Borwein-Plouffe formula, Normal Number, and Pi Alexander J. Yee & Shigeru Kondo, 5 Trillion Digits of Pi - New World Record Alexander J. Yee & Shigeru Kondo, Round 2... 10 Trillion Digits of Pi FORMULA Pi = 4*Sum_{k>=0} (-1)^k/(2k+1) [Madhava-Gregory-Leibniz, 1450-1671]. - N. J. A. Sloane, Feb 27 2013 From Johannes W. Meijer, Mar 10 2013: (Start) 2/Pi = (sqrt(2)/2) * (sqrt(2 + sqrt(2))/2) * (sqrt(2 + sqrt(2 + sqrt(2)))/2) * ... [Viete, 1593] 2/Pi = Product_{k>=1} (4*k^2-1)/(4*k^2). [Wallis, 1655] Pi = 3*sqrt(3)/4 + 24*(1/12 - Sum_{n>=2} (2*n-2)!/((n-1)!^2*(2*n-3)*(2*n+1)*2^(4*n-2))). [Newton, 1666] Pi/4 = 4*arctan(1/5) - arctan(1/239). [Machin, 1706] Pi^2/6 = 3*Sum_{n>=1} 1/(n^2*binomial(2*n,n)). [Euler, 1748] 1/Pi = (2*sqrt(2)/9801) * Sum_{n>=0} (4*n)!*(1103+26390*n)/((n!)^4*396^(4*n)). [Ramanujan, 1914] 1/Pi = 12*Sum_{n>=0} (-1)^n*(6*n)!*(13591409 + 545140134*n)/((3*n)!*(n!)^3*(64032^3)^(n+1/2)). [David and Gregory Chudnovsky, 1989] Pi = Sum_{n>=0} (1/16^n) * (4/(8*n+1) - 2/(8*n+4) - 1/(8*n+5) - 1/(8*n+6)). [Bailey-Borwein-Plouffe, 1989] (End) Pi = 4 * Sum_{k>=0} 1/(4*k+1) - 1/(4*k+3). - Alexander R. Povolotsky, Dec 25 2008 Pi = 4*sqrt(-1*(Sum_{n>=0} (i^(2*n+1))/(2*n+1))^2). - Alexander R. Povolotsky, Jan 25 2009 Pi = Integral_{x=-infinity..infinity} dx/(1+x^2). - Mats Granvik and Gary W. Adamson, Sep 23 2012 Pi - 2 = 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 - 1/36 + 1/45 + ... [Jonas Castillo Toloza, 2007], that is, Pi - 2 = Sum_{n>=1} (1/((-1)^floor((n-1)/2)*(n^2+n)/2)). - José de Jesús Camacho Medina, Jan 20 2014 Pi = 3 * Product_{t=img(r),r=(1/2+i*t) root of zeta function} (9+4*t^2)/(1+4*t^2) <=> RH is true. - Dimitris Valianatos, May 05 2016 From Ilya Gutkovskiy, Aug 07 2016: (Start) Pi = Sum_{k>=1} (3^k - 1)*zeta(k+1)/4^k. Pi = 2*Product_{k>=2} sec(Pi/2^k). Pi = 2*Integral_{x>=0} sin(x)/x dx. (End) Pi = 2^{k + 1}*arctan(sqrt(2 - a_{k - 1})/a_k) at k >= 2, where a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2). - Sanjar Abrarov, Feb 07 2017 a(n) = -10*floor(Pi*10^(-2 + n)) + floor(Pi*10^(-1 + n)) for n > 0. - Mariusz Iwaniuk, Apr 28 2017 Pi = Integral_{x = 0..2} sqrt(x/(2 - x)) dx. - Arkadiusz Wesolowski, Nov 20 2017 Pi = lim_{n->infinity} 2/n * Sum_{m=1,n} ( sqrt( (n+1)^2 - m^2 ) - sqrt( n^2 - m^2 ) ). - Dimitri Papadopoulos, May 31 2019 From Peter Bala, Oct 29 2019: (Start) Pi = Sum_{n >= 0} 2^(n+1)/( binomial(2*n,n)*(2*n + 1) ) - Euler. More generally, Pi = 4^x*x!/(2*x)! * Sum_{n >= 0} 2^(n+1)*(n+x)!*(n+2*x)!/(2*n+2*x+1)! = 2*4^x*x!^2/(2*x+1)! * hypergeom([2*x+1,1], [x+3/2], 1/2), valid for complex x not in {-1/2,-1,-3/2,-2,...}. Here, x! is shorthand notation for the function Gamma(x+1). This identity may be proved using Gauss's second summation theorem. Setting x = 3/4 and x = -1/4 (resp. x = 1/4 and x = -3/4) in the above identity leads to series representations for the constant A085565 (resp. A076390). (End) Pi = Im(log(-i^i)) = log(i^i)*(-2). - Peter Luschny, Oct 29 2019 From Amiram Eldar, Aug 15 2020: (Start) Equals 2 + Integral_{x=0..1} arccos(x)^2 dx. Equals Integral_{x=0..oo} log(1 + 1/x^2) dx. Equals Integral_{x=0..oo} log(1 + x^2)/x^2 dx. Equals Integral_{x=-oo..oo} exp(x/2)/(exp(x) + 1) dx. (End) Equals 4*(1/2)!^2 = 4*Gamma(3/2)^2. - Gary W. Adamson, Aug 23 2021 From Peter Bala, Dec 08 2021: (Start) Pi = 32*Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9))= 384*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)*(4*n^2 - 9)*(4*n^2 - 25)). More generally, it appears that for k = 1,2,3,..., Pi = 16*(2*k)!*Sum_{n >= 1} (-1)^(n+k+1)*n^2/((4*n^2 - 1)* ... *(4*n^2 - (2*k+1)^2)). Pi = 32*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^2 = 768*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2*(4*n^2 - 9)^2). More generally, it appears that for k = 0,1,2,..., Pi = 16*Catalan(k)*(2*k)!*(2*k+2)!*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2* ... *(4*n^2 - (2*k+1)^2)^2). Pi = (2^8)*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^4 = (2^17)*(3^5)*Sum_{n >= 2} (-1)^n*n^2*(n^2 - 1)/((4*n^2 - 1)^4*(4*n^2 - 9)^4) = (2^27)*(3^5)*(5^5)* Sum_{n >= 3} (-1)^(n+1)*n^2*(n^2 - 1)*(n^2 - 4)/((4*n^2 - 1)^4*(4*n^2 - 9)^4*(4*n^2 - 25)^4). (End) More generally, for odd n, Pi = (2^(n-1)/A001818((n-1)/2))*gamma(n/2)^2. - Alan Michael Gómez Calderón, Mar 11 2022 Pi = 4/phi + Sum_(n>=0) (1/phi^(12*n)) * ( 8/((12*n+3)*phi^3) + 4/((12*n+5)*phi^5) - 4/((12*n+7)*phi^7) - 8/((12*n+9)*phi^9) - 4/((12*n+11)*phi^11) + 4/((12*n+13)*phi^13) ) where phi = (1+sqrt(5))/2. - Chittaranjan Pardeshi, 16 May 2022 Pi = sqrt(3)*(27*S - 36)/24, where S = A248682. - Peter Luschny, Jul 22 2022 EXAMPLE 3.1415926535897932384626433832795028841971693993751058209749445923078164062\ 862089986280348253421170679821480865132823066470938446095505822317253594081\ 284811174502841027019385211055596446229489549303819... MAPLE Digits := 110: Pi*10^104: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 29 2019 MATHEMATICA RealDigits[ N[ Pi, 105]] [[1]] (* PROGRAM STARTS *) Clear[a, k] k = RandomInteger[{2, 10^3}]; Print["Random integer k = ", k] a[k_] := N[Nest[Sqrt[2 + #1] &, 0, k], 1000] RealDigits[N[2^(k + 1)*ArcTan[Sqrt[2 - a[k - 1]]/a[k]], 100]][[1]] (* Sanjar Abrarov, Feb 07 2017 *) Table[ResourceFunction["NthDigit"][Pi, n], {n, 1, 102}] (* Joan Ludevid, Jun 22 2022; easy to compute a(10000000)=7 with this function; requires Mathematica 12.0+ *) PROG (Macsyma) py(x) := if equal(6, 6+x^2) then 2*x else (py(x:x/3), 3*%%-4*(%%-x)^3); py(3.); py(dfloat(%)); block([bfprecision:35], py(bfloat(%))) /* Bill Gosper, Sep 09 2002 */ (PARI) { default(realprecision, 20080); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b000796.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009 (PARI) A796=[]; A000796(n)={if(n>#A796, localprec(n*6\5+29); A796=digits(Pi\.1^(precision(Pi)-3))); A796[n]} \\ NOTE: as the other programs, this returns the n-th term of the sequence, with n = 1, 2, 3, ... and not n = 1, 0, -1, -2, .... - M. F. Hasler, Jun 21 2022 (PARI) first(n)= default(realprecision, n+10); digits(floor(Pi*10^(n-1))) \\ David A. Corneth, Jun 21 2022 (Haskell)  see link: Literate Programs import Data.Char (digitToInt) a000796 n = a000796_list (n + 1) !! (n + 1) a000796_list len = map digitToInt $show$ machin' div (10 ^ 10) where    machin' = 4 * (4 * arccot 5 unity - arccot 239 unity)    unity = 10 ^ (len + 10)    arccot x unity = arccot' x unity 0 (unity div x) 1 1 where      arccot' x unity summa xpow n sign       | term == 0 = summa       | otherwise = arccot'         x unity (summa + sign * term) (xpow div x ^ 2) (n + 2) (- sign)       where term = xpow div n -- Reinhard Zumkeller, Nov 24 2012 (Haskell) See Niemeijer link and also Gibbons link. a000796 n = a000796_list !! (n-1) :: Int a000796_list = map fromInteger \$ piStream (1, 0, 1)    [(n, a*d, d) | (n, d, a) <- map (\k -> (k, 2 * k + 1, 2)) [1..]] where    piStream z xs'@(x:xs)      | lb /= approx z 4 = piStream (mult z x) xs      | otherwise = lb : piStream (mult (10, -10 * lb, 1) z) xs'      where lb = approx z 3            approx (a, b, c) n = div (a * n + b) c            mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f) -- Reinhard Zumkeller, Jul 14 2013, Jun 12 2013 (Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // Bruno Berselli, Mar 12 2013 (Python) from sympy import pi, N; print(N(pi, 1000)) # David Radcliffe, Apr 10 2019 (Python) from mpmath import mp def A000796(n):     if n >= len(A000796.str): mp.dps = n*6//5+50; A000796.str = str(mp.pi-5/mp.mpf(10)**mp.dps)     return int(A000796.str[n if n>1 else 0]) A000796.str = '' # M. F. Hasler, Jun 21 2022 CROSSREFS Cf. A001203 (continued fraction). Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), this sequence (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A224750 (b=26), A224751 (b=27), A060707 (b=60). - Jason Kimberley, Dec 06 2012 Decimal expansions of expressions involving Pi: A002388 (Pi^2), A003881 (Pi/4), A013661 (Pi^2/6), A019692 (2*Pi=tau), A019727 (sqrt(2*Pi)), A059956 (6/Pi^2), A060294 (2/Pi), A091925 (Pi^3), A092425 (Pi^4), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8), A163973 (Pi/log(2)). Cf. A001901 (Pi/2; Wallis), A002736 (Pi^2/18; Euler), A007514 (Pi), A048581 (Pi; BBP), A054387 (Pi; Newton), A092798 (Pi/2), A096954 (Pi/4; Machin), A097486 (Pi), A122214 (Pi/2), A133766 (Pi/4 - 1/2), A133767 (5/6 - Pi/4), A166107 (Pi; MGL). Cf. A248682. Sequence in context: A247385 A253214 A112602 * A212131 A114609 A271452 Adjacent sequences:  A000793 A000794 A000795 * A000797 A000798 A000799 KEYWORD cons,nonn,nice,core,easy AUTHOR EXTENSIONS Additional comments from William Rex Marshall, Apr 20 2001 STATUS approved

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