

A001113


Decimal expansion of e.
(Formerly M1727 N0684)


650



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
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OFFSET

1,1


COMMENTS

e is sometimes called Euler's number or Napier's constant.
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
Euler's constant (also known as EulerMascheroni 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 (18871920).  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
This is the expected value of the number of real numbers that are independently and uniformly chosen at random from the interval (0, 1) until their sum exceeds 1 (Bush, 1961).  Amiram Eldar, Jul 21 2020


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). 1115.
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

Mohammad K. Azarian, An Expansion of e, Problem # B765, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 4, Aug. 1995, p. 377.
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.
Eric Weisstein's World of Mathematics, e
Eric Weisstein's World of Mathematics, e Digits


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)*(1 + Sum_{n>=1} ((1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!)))^2. Robert Israel confirmed that the above formula is correct, saying: "In fact, Sum_{n=0..oo} n^j*t^n/n! = P_j(t)*exp(t) where P_0(t) = 1 and for j >= 1, P_j(t) = t (P_(j1)'(t) + P_(j1)(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
e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2.  Rok Cestnik, Jan 19 2017
The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,..., where R(n,x) is the nth row polynomial of A269953.
e = 2 + Sum_{n >= 0} (1)^n*(n+2)!/(d(n+2)*d(n+3)), where d(n) = A000166(n).
e = Sum_{n >= 0} (x^2 + (n+2)*x + n)/(n!(n + x)*(n + 1 + x)), provided x is not zero or a negative integer. (End)
Equals lim_{n > oo} (2*3*5*...*prime(n))^(1/prime(n)).  Peter Luschny, May 21 2020
e = 3  Sum_{n >= 0} 1/((n+1)^2*(n+2)^2*n!).  Peter Bala, Jan 13 2022
e = lim_{n>oo} prime(n)*(1  1/n)^prime(n).  Thomas Ordowski, Jan 31 2023
e = 1+(1/1)*(1+(1/2)*(1+(1/3)*(1+(1/4)*(1+(1/5)*(1+(1/6)*(...)))))), equivalent to the first formula.  David Ulgenes, Dec 01 2023
Equals lim_{n>oo} (1 + 1/n)^n.
Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End)


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*itfloor(10*it): od: # James A. Sellers, Feb 13 2001


MATHEMATICA



PROG

(PARI) default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(xd)*10; write("b001113.txt", n, " ", d)); \\ Harry J. Smith, Apr 15 2009
(Haskell)  See Niemeijer link.
a001113 n = a001113_list !! (n1)
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)


CROSSREFS

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


KEYWORD



AUTHOR



STATUS

approved



