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A001113 Decimal expansion of e.
(Formerly M1727 N0684)
326
2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, 0, 4, 5, 2, 3, 5, 3, 6, 0, 2, 8, 7, 4, 7, 1, 3, 5, 2, 6, 6, 2, 4, 9, 7, 7, 5, 7, 2, 4, 7, 0, 9, 3, 6, 9, 9, 9, 5, 9, 5, 7, 4, 9, 6, 6, 9, 6, 7, 6, 2, 7, 7, 2, 4, 0, 7, 6, 6, 3, 0, 3, 5, 3, 5, 4, 7, 5, 9, 4, 5, 7, 1, 3, 8, 2, 1, 7, 8, 5, 2, 5, 1, 6, 6, 4, 2, 7, 4, 2, 7, 4, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

e is sometimes called Euler's constant, also Napier's constant.

Also, decimal expansion of sinh(1)+cosh(1). - Mohammad K. Azarian, Aug 15 2006

If m and n are any integers with n > 1, then |e - m/n| > 1/(S(n)+1)!, where S(n) = A002034(n) is the smallest number such that n divides S(n)!. - Jonathan Sondow, Sep 04 2006

Abs(A000166*e-A000142) -> 0. - Seiichi Kirikami, Oct 12 2011

Euler's constant (also known as Euler-Mascheroni constant) is gamma = 0.57721... and Euler's number is e = 2.71828... . - Mohammad K. Azarian, Dec 29 2011

One of the many continued fraction expressions for e is 2+2/(2+3/(3+4/(4+5/(5+6/(6+ ... from Ramanujan (1887-1920). - Robert G. Wilson v, Jul 16 2012

e maximizes the value of x^(c/x) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. This explains why elements of A000792 are composed primarily of factors of 3, and where needed, some factors of 2. These are the two primes closest to e. - Richard R. Forberg, Oct 19 2014

There are two real solutions x to c^x = x^c when c, x > 0 and c != e, one of which is x = c, and only one real solution when c = e, where the solution is x = e. - Richard R. Forberg, Oct 22 2014

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.

E. Maor, e: The Story of a Number, Princeton Univ. Press, 1994.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 52.

G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places. Math. Tables and Other Aids to Computation 4, (1950). 11-15.

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

N. J. A. Sloane, Table of 50000 digits of e labeled from 1 to 50000 [based on the ICON Project link below]

Mohammad K. Azarian, An Expansion of e, Problem # B-765, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 4, Aug. 1995, p. 377.

Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.

Dave's Math Tables, e

X. Gourdon, Plouffe's Inverter, e to 1.250 billion digits

X. Gourdon and P. Sebah, The constant e and its computation

ICON Project, e to 50000 places

R. Nemiroff and J. Bonnell, The first 5 million digits of the number e

Remco Niemeijer, Digits Of E, programmingpraxis

J. J. O'Connor & E. F. Robertson, The number e

Simon Plouffe, A million digits

E. Sandifer, How Euler Did It, Who proved e is irrational?

D. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, Math. Comp., 23 (1969), 679-680.

Jean-Louis Sigrist, Le premier million de decimales de e. [From Lekraj Beedassy, Sep 28 2009]

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641 (article) and 114 (2007), 659 (addendum).

J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

G. Villemin's Almanach of Numbers, Constant "e"

Eric Weisstein's World of Mathematics, e

Eric Weisstein's World of Mathematics, e Digits

Eric Weisstein's World of Mathematics, Factorial Sums

Eric Weisstein's World of Mathematics, Uniform Sum Distribution

Wikipedia, E (mathematical constant)

FORMULA

e = Sum_{k >= 0} 1/k! = lim_{x -> 0} (1+x)^(1/x).

e is the unique positive root of the equation Integral_{u = 1..x} du/u = 1.

exp(1) = (16/31*(sum((1/2)^n*(1/2*n^3+1/2*n+1)/n!,n=1..infinity) +1))^2. Robert Israel confirmed that above formula is correct, saying: "In fact, sum(n^j*t^n/n!, n=0..infinity) = P_j(t)*exp(t) where P_0(t) = 1 and for j >= 1, P_j(t) = t (P_(j-1)'(t) + P_(j-1)(t)). Your sum is 1/2*P_3(1/2) + 1/2*P_1(1/2) + P_0(1/2)." - Alexander R. Povolotsky, Jan 04 2009

exp(1) = (1+ sum((1+n+n^3)/n!, n=1..infinity))/7. - Alexander R. Povolotsky, Sep 14 2011

EXAMPLE

2.71828182845904523536028747135266249775724709369995957496696762772407663...

MAPLE

Digits := 200: it := evalf((exp(1))/10, 200): for i from 1 to 200 do printf(`%d, `, floor(10*it)): it := 10*it-floor(10*it): od: # James A. Sellers, Feb 13 2001

MATHEMATICA

RealDigits[E, 10, 120][[1]] (* Harvey P. Dale, Nov 14 2011 *)

PROG

(PARI) { default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009

(Haskell) See Niemeijer link.

a001113 n = a001113_list !! (n-1)

a001113_list = eStream (1, 0, 1)

   [(n, a * d, d) | (n, d, a) <- map (\k -> (1, k, 1)) [1..]] where

   eStream z xs'@(x:xs)

     | lb /= approx z 2 = eStream (mult z x) xs

     | otherwise = lb : eStream (mult (10, -10 * lb, 1) z) xs'

     where lb = approx z 1

           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, Jun 12 2013

CROSSREFS

Cf. A002034, A122214, A122215, A122216, A122217, A122416, A122417.

Expansion of e in base b: A004593 (b=2), A004594 (b=3), A004595 (b=4), A004596 (b=5), A004597 (b=6), A004598 (b=7), A004599 (b=8), A004600 (b=9), this sequence (b=10), A170873 (b=16). - Jason Kimberley, Dec 05 2012

Sequence in context: A170936 A111714 A060302 * A248685 A182587 A248677

Adjacent sequences:  A001110 A001111 A001112 * A001114 A001115 A001116

KEYWORD

nonn,cons,nice,core,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 21 17:16 EST 2014. Contains 252324 sequences.