A061382 -id:A061382 - cp's OEIS frontend

A061360 Decimal expansion of e/Pi.

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

Offset: 0

Jason Earls, Jun 07 2001

			0.86525597943226508721777478964608961742874...

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

Cf. A000796, A001113, A007526, A061382.

RealDigits[N[E/Pi,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
RealDigits[E/Pi,10,120][[1]] (* Harvey P. Dale, Sep 29 2023 *)

{ default(realprecision, 20080); x=10*exp(1)/Pi; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b061360.txt", n, " ", d)); } \\ Harry J. Smith, Jul 21 2009

Equals -1 + Product_{n>=0} (1 + 1/(A007526(n) + n!*Pi)). - David Ulgenes, Sep 21 2023

Edited by N. J. A. Sloane, Sep 18 2008 at the suggestion of R. J. Mathar

A073244 Decimal expansion of Pi - e.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 21 2002

			0.42331082513074800310235591192...

Cf. A059742 (Pi+e), A000796 (Pi), A001113 (e), A019609 (Pi*e), A061382 (Pi/e), A061360 (e/Pi), A039661 (e^Pi), A059850 (Pi^e), A073233 (Pi^Pi), A073226 (e^e), A049006 (i^i = e^(-Pi/2)).

Cf. A110564 for continued fraction for Pi - e.

RealDigits[N[Pi-E,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI
```
Pi-exp(1)
```

A092034 Decimal expansion of (e/Pi)^2.

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Mar 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A061382.

RealDigits[(E/Pi)^2, 10, 100][[1]] (* Vincenzo Librandi, Mar 01 2013 *)

A092035 Decimal expansion of (Pi/e)^2.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 26 2004

			1.33570570...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A061382, A100074.

RealDigits[(Pi/E)^2, 10, 100][[1]] (* Vincenzo Librandi, Mar 01 2013 *)

Conjecture: Equals 12*Sum_{k >= 1} ((-1)^(k+1)/k^2) * cos(sqrt(2*k^2*Pi^2/3 - 4)). Cf. A100074. - Peter Bala, Jun 20 2022

A032739 a(n) = floor( (Pi/e)^n ).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 37, 43, 49, 57, 66, 76, 88, 102, 118, 137, 158, 183, 211, 244, 282, 326, 377, 436, 504, 582, 673, 778, 899, 1040, 1202, 1389, 1605, 1855, 2144, 2478, 2864, 3310, 3826, 4421, 5110, 5906

Offset: 0

Patrick De Geest, May 15 1998

Cf. A061382 (Pi/e).

Floor[(Pi/E)^Range[0,60]] (* Harvey P. Dale, Sep 30 2011 *)

a(n) = floor( (Pi/exp(1))^n ); \\ Michel Marcus, Jun 09 2020

a(0)=1 prepended by Georg Fischer, Jun 09 2020

A096414 Decimal expansion of (Pi/e)^(1/2).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Aug 07 2004

			1.07504760349992023872275586024820851177...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A000796, A001113, A061382, A257530.

evalf(Pi/exp(1)); # R. J. Mathar, Oct 03 2011

RealDigits[N[Sqrt[Pi/E], 120]] (* Michael De Vlieger, Apr 30 2015 *)

a = sqrt(Pi/exp(1)) \\ Stanislav Sykora, Apr 28 2015

Equals the square root of A061382.

Equals integral[-inf..+inf](exp(-x^2)*cos(k*x)) = sqrt(Pi/exp(k^2/2)), for k = sqrt(2). - Stanislav Sykora, Apr 28 2015

A283743 Decimal expansion of Ei(1)/e, where Ei is the exponential integral function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 15 2017

Can be considered the value of the divergent series -0! - 1! - 2! - ... ; see Lagarias reference Section 2.5. - Harry Richman, Jun 14 2020.

			0.6971748832350660687654786819195515953171754309543695173200548...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, equation 44:5:10 at page 426.

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.
Eric Weisstein's World of Mathematics, Exponential Integral.
Eric Weisstein's World of Mathematics, Subfactorial.

Cf. A000166 (subfactorials), A061382 (Pi/e, the imaginary part of subfactorial(-1)), A091725 (Ei(1)), A073003 (-exp(1)*Ei(-1)).

RealDigits[ExpIntegralEi[1]/E, 10, 102][[1]]

real(-eint1(-1)/exp(1)) \\ Michel Marcus, Jun 15 2020

Equals Re(subfactorial(-1)) = Re(Gamma(0,-1)/e).

Equals Sum_{k=1..oo} (-1)^k*psi(k)/Gamma(k), where psi denotes the digamma function (see Spanier and Oldham). - Stefano Spezia, Jan 04 2025

A061666 Continued fraction for Pi/e.

Original entry on oeis.org

1, 6, 2, 2, 1, 2, 6, 8, 2, 1, 1, 1, 4, 3, 1, 1, 66, 2, 1, 1, 2, 4, 2, 7, 46, 10, 2, 1, 18, 1, 23, 10, 14, 1, 1, 4, 3, 1, 6, 2, 5, 1, 1, 1, 18, 355, 1, 1, 5, 3, 4, 1, 28, 5, 2, 1, 3, 16, 3, 25, 4, 7, 1, 11, 2, 2, 1, 9, 2, 31, 1, 20, 1, 2, 1, 1, 6, 13, 1, 3, 1, 3, 4, 125, 1, 7, 2, 2, 1, 3, 5, 1, 1, 6, 1

Offset: 0

Jason Earls, Jun 16 2001

			1.155727349790921717910093183... = 1 + 1/(6 + 1/(2 + 1/(2 + 1/(1 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..19999

Cf. A061382 (decimal expansion).

ContinuedFraction[Pi/E,100] (* Harvey P. Dale, Oct 31 2011 *)

PARI
```
contfrac(Pi/exp(1))
```

allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi/exp(1)); for (n=1, 20000, write("b061666.txt", n-1, " ", x[n])); \\ Harry J. Smith, Jul 25 2009

Extra comma removed from terms by Harry J. Smith, Jul 25 2009

Offset changed by Andrew Howroyd, Aug 04 2024

A092156 Decimal expansion of (Pi/e)^3.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 31 2004

			1.54371161691

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A061382, A092035.

RealDigits[(Pi/E)^3, 10, 100][[1]] (* Vincenzo Librandi, Mar 01 2013 *)

A092157 Decimal expansion of (Pi/e)^4.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 31 2004

			1.7841097358586

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A061382, A092035.

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

cp's OEIS Frontend

A061360 Decimal expansion of e/Pi.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A073244 Decimal expansion of Pi - e.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A092034 Decimal expansion of (e/Pi)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A092035 Decimal expansion of (Pi/e)^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A032739 a(n) = floor( (Pi/e)^n ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A096414 Decimal expansion of (Pi/e)^(1/2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A283743 Decimal expansion of Ei(1)/e, where Ei is the exponential integral function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A061666 Continued fraction for Pi/e.

Views

Author

Keywords

Examples

Links