

A001113


Decimal expansion of e.
(Formerly M1727 N0684)


589



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.
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
Limit_{n>infinity} A000166(n)*e  A000142(n) = 0.  Seiichi Kirikami, Oct 12 2011
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

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 # B765, 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.
L. E. Bush, The William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 68, No. 1 (1961), pp. 1833, problem 3.
Ed Copeland and Brady Haran, A proof that e is irrational, Numberphile video (2021).
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
Michael Penn, e is irrational, YouTube video, 2020.
Simon Plouffe, A million digits
G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places, Pi, A Source book, pp 277281, 2000.
E. Sandifer, How Euler Did It, Who proved e is irrational?, MAA Online (2006)
D. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, Math. Comp., 23 (1969), 679680.
JeanLouis Sigrist, Le premier million de décimales de e
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637641 (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
Eric Weisstein's World of Mathematics, An amazing pandigital approximation to e that is correct to 18457734525360901453873570 decimal digits
Wikipedia, E (mathematical constant)
Index entries for "core" sequences
Index entries for transcendental numbers


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^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_(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
exp(1) = (1 + Sum_{n>=1} ((1+n+n^3)/n!))/7.  Alexander R. Povolotsky, Sep 14 2011
e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2.  Rok Cestnik, Jan 19 2017
a(n) = 10*floor(exp(1)*10^(2 + n)) + floor(exp(1)*10^(1 + n)) for n > 0.  Mariusz Iwaniuk, Apr 28 2017
From Peter Bala, Nov 13 2019: (Start)
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 > infinity} (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


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

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=(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)
 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
Powers e^k: A092578 (k = 7), A092577 (k = 6), A092560 (k = 5), A092553  A092555 (k = 2 to 4), A068985 (k = 1), A072334 (k = 2), A091933 (k = 3), A092426 (k = 4), A092511  A092513 (k = 5 to 7).
Sequence in context: A170936 A111714 A060302 * A248685 A182587 A248677
Adjacent sequences: A001110 A001111 A001112 * A001114 A001115 A001116


KEYWORD

nonn,cons,nice,core


AUTHOR

N. J. A. Sloane


STATUS

approved



