A092512 -id:A092512 - cp's OEIS frontend

A072334 Decimal expansion of e^2.

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)

A092426 Decimal expansion of e^4.

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Mar 22 2004

			54.598150033144239078110261202860878402790737038614....

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

Cf. A001113, A072334, A091933, A092511, A092512, A092513.

Cf. A058283 (continued fraction).

Digits:=100: evalf(exp(4)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[E^4, 10, 100][[1]] (* Alonso del Arte, Aug 31 2014 *)

default(realprecision, 20080); x=exp(1)^4/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b092426.txt", n, " ", d)); \\ Harry J. Smith, Jun 22 2009

From Peter Bala, Jan 12 2022: (Start)
e^4 = 45 + 2*Sum_{n >= 0} 4^(n+4)/((n+4)^2*(n+5)^2*n!).
45/e^4 = 1 - 3*Sum_{n >= 0} (-4)^(n+3)*n^2/(n+4)!. (End)
Largest solution to sqrt(log(x)) = log(sqrt(x)), the other solution being x = 1. - Andrea Pinos, Jan 23 2024

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)

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)

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

A277117 Decimal expansion of e^6/(Pi^5+Pi^4), where e = exp(1).

Original entry on oeis.org

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

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

			1.0000000438081076299476374320122890058190409215306036959233520...

D. Wilson, pers. comm.

Eric Weisstein's World of Mathematics, Almost Integer
Wikipedia, Near-identities involving e and pi.

Cf. A000796, A001113, A092425, A092731, A092512, A239606.

RealDigits[E^6/(Pi^5+Pi^4),10,120][[1]] (* Harvey P. Dale, Apr 07 2019 *)

exp(6)/(Pi^5+Pi^4) \\ Michel Marcus, Oct 02 2016

Equals A092512 /(A092731 + A092425).

A376994 Decimal expansion of Pi^4 + Pi^5.

Original entry on oeis.org

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

Offset: 3

Stefano Spezia, Oct 12 2024

This constant is very close to e^6 = A092512.

			403.4287758192838904991816427321407177300282923...

Index entries for transcendental numbers.

Cf. A092425, A092512, A092731.

Mathematica
```
RealDigits[Pi^4+Pi^5,10,100][[1]]
```

cp's OEIS Frontend

A072334 Decimal expansion of e^2.

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A091933 Decimal expansion of e^3.

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A092426 Decimal expansion of e^4.

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A092511 Decimal expansion of e^5.

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A092513 Decimal expansion of e^7.

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

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A277117 Decimal expansion of e^6/(Pi^5+Pi^4), where e = exp(1).

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