A092426 -id:A092426 - cp's OEIS frontend

A001113 Decimal expansion of e.

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

Offset: 1

N. J. A. Sloane

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 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
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

			2.71828182845904523536028747135266249775724709369995957496696762772407663...

Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 400.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 250-256.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers, p. 85.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 1 and 2, equations 1:7:4, 2:5:4 at pages 13, 20.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.

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.
L. E. Bush, The William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 68, No. 1 (1961), pp. 18-33, 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
Brady Haran and James Grime, Incredible Formula, Numberphile YouTube video, 2016.
ICON Project, e to 50000 places
Roger Mansuy, Un intrigant poème... mathématique, Images des Mathématiques, CNRS, 2023. In French.
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 277-281, 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), 679-680.
Jean-Louis Sigrist, Le premier million de décimales de e.
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 339.
Jonathan 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).
Jonathan 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, e Approximations
Wikipedia, E (mathematical constant)
Index entries for "core" sequences
Index entries for transcendental numbers

Cf. A002034, A003417 (continued fraction), A073229, 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).

-- 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

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 Sellers, Feb 13 2001

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

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_(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_{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
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 n-th 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
From Michal Paulovic, Dec 12 2023: (Start)
Equals lim_{n->oo} (1 + 1/n)^n.
Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End)
Equals Product_{k>=1} (1 + 1/k) * (1 - 1/(k + 1)^2)^k. - Antonio Graciá Llorente, May 14 2024
Equals lim_{n->oo} Product_{k=1..n} (n^2 + k)/(n^2 - k) (see Finch). - Stefano Spezia, Oct 19 2024
e ~ (1 + 9^((-4)^(7*6)))^(3^(2^85)), correct to more than 18*10^24 digits (Richard Sabey, 2004); see Haran and Grime link. - Paolo Xausa, Dec 21 2024.

A072334 Decimal expansion of e^2.

Original entry on oeis.org

7, 3, 8, 9, 0, 5, 6, 0, 9, 8, 9, 3, 0, 6, 5, 0, 2, 2, 7, 2, 3, 0, 4, 2, 7, 4, 6, 0, 5, 7, 5, 0, 0, 7, 8, 1, 3, 1, 8, 0, 3, 1, 5, 5, 7, 0, 5, 5, 1, 8, 4, 7, 3, 2, 4, 0, 8, 7, 1, 2, 7, 8, 2, 2, 5, 2, 2, 5, 7, 3, 7, 9, 6, 0, 7, 9, 0, 5, 7, 7, 6, 3, 3, 8, 4, 3, 1, 2, 4, 8, 5, 0, 7, 9, 1, 2, 1, 7, 9

Offset: 1

N. J. A. Sloane, Jul 15 2002

Also where x^(1/sqrt(x)) is a maximum. - Robert G. Wilson v, Oct 22 2014

			7.389056098930650...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 1.4, pages 2 and 28-29.

Harry J. Smith, Table of n, a(n) for n = 1..20000
John Cosgrave, e^2 is irrational.
Index entries for transcendental numbers.

Cf. A001204 (continued fraction).

Cf. A001113, A091933, A092426, A092511, A092512, A092513.

SetDefaultRealField(RealField(100)); Exp(1)^2; // Vincenzo Librandi, Apr 05 2020

RealDigits[E^2, 10, 100][[1]] (* Vincenzo Librandi, Apr 05 2020 *)

default(realprecision, 20080); x=exp(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b072334.txt", n, " ", d)); \\ Harry J. Smith, Apr 30 2009

Equals Sum_{n>=0} Sum_{k>=0} 1/(n!*k!). - Fredrik Johansson, Apr 21 2006
Equals Sum_{n>=0} 2^n/n!. - Daniel Hoyt Nov 20 2020
From Peter Bala, Jan 13 2022: (Start)
e^2 = Sum_{n >= 0} 2^n/n!. Faster converging series include
e^2 = 8*Sum_{n >= 0} 2^n/(p(n-1)*p(n)*n!), where p(n) = n^2 - n + 2 and
e^2 = -48*Sum_{n >= 0} 2^n/(q(n-1)*q(n)*n!), where q(n) = n^3 + 5*n - 2.
e^2 = 7 + Sum_{n >= 0} 2^(n+3)/((n+2)^2*(n+3)^2*n!) and
7/e^2 = 1 - Sum_{n >= 0} (-2)^(n+1)*n^2/(n+2)!.
e^2 = 7 + 2/(5  + 1/(7 + 1/(9  + 1/(11 + ...)))) (follows from the fact that A004273 is the continued fraction expansion of tanh(1) = (e^2 - 1)/ (e^2 + 1)). Cf. A001204. (End)
Equals lim_{n->oo} (Sum_{k=1..n} 1/binomial(n,k)^x)^(n^x), for all real x > 1/2 (Furdui, 2013). - Amiram Eldar, Mar 26 2022

A091933 Decimal expansion of e^3.

Original entry on oeis.org

2, 0, 0, 8, 5, 5, 3, 6, 9, 2, 3, 1, 8, 7, 6, 6, 7, 7, 4, 0, 9, 2, 8, 5, 2, 9, 6, 5, 4, 5, 8, 1, 7, 1, 7, 8, 9, 6, 9, 8, 7, 9, 0, 7, 8, 3, 8, 5, 5, 4, 1, 5, 0, 1, 4, 4, 3, 7, 8, 9, 3, 4, 2, 2, 9, 6, 9, 8, 8, 4, 5, 8, 7, 8, 0, 9, 1, 9, 7, 3, 7, 3, 1, 2, 0, 4, 4, 9, 7, 1, 6, 0, 2, 5, 3, 0, 1, 7, 7, 0

Offset: 2

Mohammad K. Azarian, Mar 16 2004

Also where x^(1/x^(1/3)) is a maximum. - Robert G. Wilson v, Oct 22 2014

			exp(3) = e^3 = 20.0855369231876677409285296545817178969879... - _Harry J. Smith_, Apr 30 2009

Harry J. Smith, Table of n, a(n) for n = 2..20000
Index entries for transcendental numbers

Cf. A001113, A072334, A092426, A092511, A092512, A092513.

Digits:=100: evalf(exp(3)); # Wesley Ivan Hurt, Jul 07 2014

RealDigits[E^3, 10, 100][[1]] (* Alonso del Arte, Jul 07 2014 *)

default(realprecision, 20080); x=exp(3)/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b091933.txt", n, " ", d)); \\ Harry J. Smith, Apr 30 2009

From Peter Bala, Jan 12 2022: (Start)
e^3 = Sum_{n >= 0} 3^n/n!. Faster converging series include
e^3 = 18*Sum_{n >= 0} 3^n/(p(n-1)*p(n)*n!), where p(n) = n^2 - 3*n + 5 and
e^3 = -162*Sum_{n >= 0} 3^n/(q(n-2)*q(n-1)*n!), where q(n) = n^3 + 8*n - 3.
e^3 = 22 - Sum_{n >= 0} 3^(n+4)/((n+3)^2*(n+4)^2*n!) and
22/e^3 = 1 - 2*Sum_{n >= 0} (-3)^(n+2)*n^2/(n+3)!.
e^3 = lim_{n -> oo} f(n+2)*f(n)/(n^2*f(n+1)^2), where f(n) = n^(n^2). Compare with e = lim_{n -> oo} g(n+1)/(n*g(n)), where g(n) = n^n. (End)

A092042 Decimal expansion of e^(1/4).

Original entry on oeis.org

1, 2, 8, 4, 0, 2, 5, 4, 1, 6, 6, 8, 7, 7, 4, 1, 4, 8, 4, 0, 7, 3, 4, 2, 0, 5, 6, 8, 0, 6, 2, 4, 3, 6, 4, 5, 8, 3, 3, 6, 2, 8, 0, 8, 6, 5, 2, 8, 1, 4, 6, 3, 0, 8, 9, 2, 1, 7, 5, 0, 7, 2, 9, 6, 8, 7, 2, 2, 0, 7, 7, 6, 5, 8, 6, 7, 2, 3, 8, 0, 0, 2, 7, 5, 3, 3, 0, 6, 4, 1, 9, 4, 3, 9, 5, 5, 3, 5, 6, 8

Offset: 1

Mohammad K. Azarian, Mar 27 2004

e^(1/4) is also the integral from 0 to infinity of e^(-x) * I_0(sqrt(x)), where I_0(z) is a modified Bessel function. - Jean-François Alcover, Mar 10 2011

e^(1/4) maximizes the value of x^(c/(x^4)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

			1.28402541668774148407342056806243645833....

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
D. M. Bătinetu-Giurgiu, Problem 4133, Crux Mathematicorum, Vol. 42, No. 4 (2016), p. 174; Solution to Problem 4133, ibid., Vol. 43, No. 4 (2017), pp. 167-169.
Index entries for transcendental numbers

Cf. A001113, A019774, A057863, A078688, A092426.

evalf(exp(1/4)); # Muniru A Asiru, Aug 16 2018

RealDigits[(E)^(1/4), 10, 100][[1]] (* Vincenzo Librandi, Mar 01 2013 *)

PARI
```
exp(1/4) \\ Michel Marcus, Jan 19 2017
```

e^(1/4) = 1/2*( 1 +(5 +(9 +(13 +...)/12)/8)/4 ) = 1 +(1 +(1 +(1 +...)/12)/8)/4. - Rok Cestnik, Jan 19 2017
Equals lim_{n->oo} ((2*n-1)!!)^(1/(2*n))/A057863(n)^(1/n^2) (Bătinetu-Giurgiu, 2016). - Amiram Eldar, Apr 10 2022
Equals (Integral_{x=1..oo} 1/(x*log(x)^log(log(x))) dx)/sqrt(Pi). - Kritsada Moomuang, Jun 03 2025

A092511 Decimal expansion of e^5.

Original entry on oeis.org

1, 4, 8, 4, 1, 3, 1, 5, 9, 1, 0, 2, 5, 7, 6, 6, 0, 3, 4, 2, 1, 1, 1, 5, 5, 8, 0, 0, 4, 0, 5, 5, 2, 2, 7, 9, 6, 2, 3, 4, 8, 7, 6, 6, 7, 5, 9, 3, 8, 7, 8, 9, 8, 9, 0, 4, 6, 7, 5, 2, 8, 4, 5, 1, 1, 0, 9, 1, 2, 0, 6, 4, 8, 2, 0, 9, 5, 8, 5, 7, 6, 0, 7, 9, 6, 8, 8, 4, 0, 9, 4, 5, 9, 8, 9, 9, 0, 2, 1, 1, 4, 1, 2, 9, 2

Offset: 3

Mohammad K. Azarian, Apr 05 2004

			148.413159102

Index entries for transcendental numbers

Cf. A001113, A072334, A091933, A092426, A092512, A092513.

RealDigits[E^5,10,120][[1]] (* Harvey P. Dale, May 31 2020 *)

exp(5) \\ Charles R Greathouse IV, Oct 02 2022

From Peter Bala, Jan 12 2022: (Start)
e^5 = 456 - Sum_{n >= 0} 5^(n+6)/((n+5)^2*(n+6)^2*n!).
456/e^5 = 1 - 4!*Sum_{n >=0} (-5)^(n+4)*n^2/(n+5)!. (End)

A092512 Decimal expansion of e^6.

Original entry on oeis.org

4, 0, 3, 4, 2, 8, 7, 9, 3, 4, 9, 2, 7, 3, 5, 1, 2, 2, 6, 0, 8, 3, 8, 7, 1, 8, 0, 5, 4, 3, 3, 8, 8, 2, 7, 9, 6, 0, 5, 8, 9, 9, 8, 9, 7, 3, 5, 7, 1, 2, 9, 2, 0, 2, 6, 1, 3, 9, 6, 7, 1, 8, 8, 3, 2, 5, 1, 5, 1, 1, 8, 0, 6, 3, 3, 9, 9, 3, 4, 9, 8, 3, 0, 5, 1, 7, 8, 8, 8, 6, 6, 5, 1, 2, 1, 2, 6, 6, 4, 7, 6, 8, 3, 7, 5

Offset: 3

Mohammad K. Azarian, Apr 05 2004

Curiously, e^6 = 403.428793... is very close to Pi^4 + Pi^5 = 403.428775... - Jean-François Alcover, Jul 04 2013

			403.428793492735122...

Index entries for transcendental numbers

Cf. A001113, A072334, A091933, A092426, A092511, A092513.

RealDigits[E^6, 10, 108][[1]] (* Alonso del Arte, Oct 05 2014 *)

exp(6) \\ Charles R Greathouse IV, Oct 02 2022

From Peter Bala, Jan 12 2022: (Start)
e^6 = 115 + (1/4)*Sum_{n >= 0} 6^(n+6)/((n+6)^2*(n+7)^2*n!).
115/e^6 = 1 - 5*Sum_{n >= 0} (-6)^(n+5)*n^2/(n+6)!. (End)

A092513 Decimal expansion of e^7.

Original entry on oeis.org

1, 0, 9, 6, 6, 3, 3, 1, 5, 8, 4, 2, 8, 4, 5, 8, 5, 9, 9, 2, 6, 3, 7, 2, 0, 2, 3, 8, 2, 8, 8, 1, 2, 1, 4, 3, 2, 4, 4, 2, 2, 1, 9, 1, 3, 4, 8, 3, 3, 6, 1, 3, 1, 4, 3, 7, 8, 2, 7, 3, 9, 2, 4, 0, 7, 7, 6, 1, 2, 1, 7, 6, 9, 3, 3, 1, 2, 3, 3, 1, 2, 9, 0, 2, 2, 4, 7, 8, 5, 6, 8, 7, 8, 7, 2, 4, 9, 8, 4, 3, 8, 8, 4, 2, 0

Offset: 4

Mohammad K. Azarian, Apr 05 2004

			1096.6331584284585992...

Index entries for transcendental numbers

Cf. A001113, A072334, A091933, A092426, A092511, A092512.

RealDigits[E^7,10,120][[1]] (* Harvey P. Dale, Dec 06 2012 *)

exp(7) \\ Charles R Greathouse IV, Oct 02 2022

From Peter Bala, Jan 12 2022: (Start)
e^7 = (1/167)*Sum_{n >= 0} 7^(n+8)/((n+7)^2*(n+8)^2*n!) - 19440/167.
19440/e^7 = 7!*Sum_{n >= 0} (-7)^(n+5)*n^2/(n+7)! - 167. (End)

A058283 Continued fraction for e^4.

Original entry on oeis.org

54, 1, 1, 2, 21, 4, 1, 1, 57, 2, 1, 9, 1, 1, 13, 6, 2, 20, 2, 4, 1, 17, 1, 3, 2, 7, 2, 1, 1, 1, 1, 3, 2, 1, 11, 4, 2, 1, 3, 2, 1, 7, 1, 7, 1, 12, 2, 8, 7, 6, 1, 2, 1, 7, 2, 37, 2, 3, 66, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 2, 10, 18, 2, 19, 4, 2, 1, 9, 1, 9, 2, 1, 1, 1, 1, 4

Offset: 0

Robert G. Wilson v, Dec 07 2000

			54.598150033144239078110261... = 54 + 1/(1 + 1/(1 + 1/(2 + 1/(21 + ...)))). - _Harry J. Smith_, Jun 22 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A092426 Decimal expansion. - Harry J. Smith, Jun 22 2009

Mathematica
```
ContinuedFraction[ E^4, 100]
```

default(realprecision,1000); contfrac((exp(1))^4)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(1)^4); for (n=0, 20000, write("b058283.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 22 2009

More terms from R. K. Guy, Jun 15 2001

A254528 Number of decimal digits in the integer part of e^n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32, 33, 33

Offset: 0

Robert G. Wilson v, Feb 01 2015

			e^10 = 22026.46579480671..., so a(10) = 5.

Benedict W. J. Irwin, Table of n, a(n) for n = 0..10000

Cf. A000149, A055642.

Cf. A001113, A072334, A091933, A092426, A092511, A092512, A092513 (see their offsets).

f[n_] := 1 + Floor@ Log10@ Exp@ n; Array[f, 75, 0]
Table[Sum[DigitCount[Floor[Exp[1]^k]][[n]], {n, 1, 10}], {k, 0, 150}] (* Benedict W. J. Irwin, Apr 13 2016 *)
IntegerLength[Floor[E^Range[0,80]]] (* Harvey P. Dale, Aug 28 2017 *)

a(n) = localprec(n+1); #Str(floor(exp(n))); \\ Michel Marcus, Dec 05 2020

a(n) = A055642(A000149(n)). - Amiram Eldar, May 25 2024

A309887 Coefficients of 4-adic expansion of exp(4).

Original entry on oeis.org

1, 3, 0, 1, 1, 0, 0, 1, 2, 3, 0, 2, 0, 2, 3, 0, 2, 0, 3, 1, 2, 2, 1, 0, 2, 2, 3, 0, 3, 2, 2, 3, 0, 2, 1, 0, 0, 0, 2, 3, 0, 2, 0, 0, 3, 3, 3, 1, 3, 0, 2, 2, 3, 3, 3, 1, 0, 3, 1, 2, 0, 0, 2, 3, 3, 1, 1, 0, 2, 1, 3, 2, 1, 1, 0, 1, 2, 3, 2, 3, 1, 1, 2, 2, 0, 3, 3, 0, 1, 1, 1, 2, 0, 1, 3, 0, 2, 2, 0, 3

Offset: 0

Seiichi Manyama, Aug 21 2019

Robert Israel, Table of n, a(n) for n = 0..9999

k-adic expansion of exp(k): A317675 (k=3), this sequence (k=4), A309888 (k=5).

Cf. A092426.

op([1,3],padic:-evalp(exp(4),4,100)); # Robert Israel, Aug 02 2020

N=100; Vecrev(digits(lift(exp(4+O(2^(2*N)))), 4), N)

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