A059742 -id:A059742 - cp's OEIS frontend

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A073244 Decimal expansion of Pi - e.

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)
```

A058651 Continued fraction for Pi + e.

Original entry on oeis.org

5, 1, 6, 7, 3, 21, 2, 1, 2, 2, 1, 1, 2, 3, 3, 2, 5, 2, 1, 1, 1, 1, 3, 1, 8, 4, 4, 1, 1, 1, 1, 8, 1, 4, 1, 5, 1, 1, 1, 2, 4, 3, 2, 1, 1, 2, 1, 10, 1, 4, 1, 2, 1, 12, 1, 8, 2, 7, 39, 365, 2, 15, 2, 25, 1, 2, 5, 3, 3, 9, 3, 1, 1, 9, 1, 1, 47, 1, 1, 18, 1, 1, 2, 6, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 4, 1, 6, 37

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Dec 26 2000

The question of the transcendence of the number Pi + e is still open.

			a(1) = 5 because Pi + e = 5.859874482048838473822930854632165381954416493075065395941912220031...
5.859874482048838473822930854... = 5 + 1/(1 + 1/(6 + 1/(7 + 1/(3 + ...)))). - _Harry J. Smith_, May 31 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A001203, A003417.

Cf. A059742 (decimal expansion).

PARI
```
\p 500; contfrac(Pi+exp(1))
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi+exp(1)); for (n=1, 20000, write("b058651.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 31 2009

More terms from Jason Earls, Jun 28 2001

Offset changed by Andrew Howroyd, Aug 04 2024

A094001 Decimal expansion of (Pi + e)^Pi.

Original entry on oeis.org

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

Offset: 3

Mohammad K. Azarian, May 30 2004

			258.4593719667472887161071021684269...

Harry J. Smith, Table of n, a(n) for n = 3..20000

Cf. A000796, A001113, A059742.

Cf. A161683 (continued fraction). - Harry J. Smith, Jun 16 2009

RealDigits[(Pi + E)^Pi, 10, 125][[1]] (* G. C. Greubel, Dec 27 2021 *)

{ default(realprecision, 20080); x=(Pi+exp(1))^Pi/100; for (n=3, 20000, d=floor(x); x=(x-d)*10; write("b094001.txt", n, " ", d)); } \\ Harry J. Smith, Jun 16 2009

numerical_approx((pi+e)^pi, digits=125) # G. C. Greubel, Dec 27 2021

A121831 a(n) = floor((Pi + e)^n).

Original entry on oeis.org

1, 5, 34, 201, 1179, 6909, 40488, 237256, 1390293, 8146945, 47740079, 279750872, 1639304997, 9606121523, 56290666385, 329856239532, 1932916160781, 11326646086501, 66372724369487, 388935833836819, 2279115167854766

Offset: 0

Mohammad K. Azarian, Aug 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A059742 (Pi+e), A121282.

Table[Floor[(Pi + E)^n], {n, 0, 80}] (* Vincenzo Librandi, Feb 21 2013 *)

Offset changed by Georg Fischer, Sep 02 2022

A121899 a(n) = ceiling((Pi + e)*sqrt(n)).

Original entry on oeis.org

0, 6, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 42, 43, 43, 44, 44, 44, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A059742 (Pi+e), A121854, A121855, A121856, A121857.

C := ComplexField(); [Ceiling((Exp(1) + Pi(C))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017

Mathematica

Table[Ceiling[(Pi + E) Sqrt[n]], {n, 0, 70}] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil((exp(1)+Pi)*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121901 a(n) = floor((Pi + e)*sqrt(n)).

Original entry on oeis.org

0, 5, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 41, 42, 42, 43, 43, 43, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48

Offset: 0

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Author

Mohammad K. Azarian, Sep 01 2006

Keywords

nonn
easy

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Crossrefs

Cf. A059742 (Pi+e), A121854, A121855, A121856, A121857, A121900, A121902.

Programs

Magma

C := ComplexField(); [Floor((Exp(1) + Pi(C))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017

Mathematica

Table[Floor[(Pi + E) Sqrt[n]], {n, 0, 70}] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(floor((exp(1)+Pi)*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

A291854 Decimal expansion of the "Pi AND e" (Piande) constant.

Original entry on oeis.org

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

Offset: 1

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Author

Alois P. Heinz, Sep 13 2017

Keywords

nonn
cons
base

Comments

Pi_2 = 11.00100100001111110... -> 3.1415926535897...

e_2 = 10.10110111111000010... -> 2.7182818284590...

Pi_2 AND e_2 = 10.00100100001000010... -> 2.1411323548533...

Examples

2.1411323548533155483334288212398570571182864149673143672536880794...

Crossrefs

Cf. A000796, A001113, A004593, A004601, A019609, A059742, A114916, A228239, A231325, A292101, A292107, A291858.

Formula

Equals A059742 - A291858 = A291858 - A114916.

A291858 Decimal expansion of the "Pi OR e" (Piore) constant.

Original entry on oeis.org

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

Offset: 1

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Author

Alois P. Heinz, Sep 13 2017

Keywords

nonn
cons
base

Comments

Pi_2 = 11.00100100001111110... -> 3.1415926535897...

e_2 = 10.10110111111000010... -> 2.7182818284590...

Pi_2 OR e_2 = 11.10110111111111110... -> 3.7187421271955...

Examples

3.71874212719552292548950203339230832483613007810775102868822414063...

Crossrefs

Cf. A000796, A001113, A004593, A004601, A019609, A059742, A114916, A228239, A231325, A292101, A292107, A291854.

Formula

Equals A059742 - A291854 = A291854 + A114916.

A094244 Decimal expansion of 1/sqrt(Pi + e).

Original entry on oeis.org

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

Offset: 0

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Author

Mohammad K. Azarian, May 30 2004

Keywords

cons
nonn

Examples

0.413100616602795399...

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A001113, A000796, A059742.

Programs

Mathematica

RealDigits[N[1/Sqrt[Pi + E], 100]][[1]] (* Vincenzo Librandi, Aug 09 2015 *)

PARI

1/sqrt(Pi + exp(1)) \\ G. C. Greubel, Jan 13 2017

Extensions

Offset corrected by Mohammad K. Azarian, Dec 11 2008

A352396 Integer part of e[n]Pi, where [n] indicates hyper-n, e = 2.718281828459045..., and Pi = 3.141592653589793... (using H. Kneser's proposal for n > 3).

Original entry on oeis.org

4, 5, 8, 23, 37149801960

Offset: 0

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Author

Marco Ripà, Apr 08 2022

Keywords

nonn
more
hard

Comments

The first term of this sequence is given by floor(e[0]Pi) = floor(Pi + 1) = floor(4.14159) = 4, which is the integer part of "e zeration Pi". In general, zeration is not a commutative arithmetic operation, while floor(e[1]Pi) = floor(Pi + e) = floor(5.85987) = 5 and floor(e[2]Pi) = floor(Pi * e) = floor(8.53973) = 8 hold since e[1]Pi = Pi[1]e and e[2]Pi = Pi[2]e.

If n = 3, then floor(e[3]Pi) = floor(e^Pi) = floor(23.14069) = 23 (if n > 2, then hyper-n is not characterized by the commutative property anymore, even if we can find fascinating examples as 4[3]2 = 2[3]4 = 16).

Now, tetration can be extended to complex bases as described in the Paulsen reference and the corresponding term of the present sequence can be found using his online calculator (see Links), so we have that floor(e[4]Pi) = floor(37149801960.55) = 37149801960. An easy proof that 37149801960.55999 > e^^Pi > 37149801960.55 follows from the chain of inequalities 37149801960.5569855999 > |37149801960.5569855 + 5.9249049902894650649*10^(-11)| > e^^Pi > |37149801960.556985498 + 5.9249049902894650647*10^(-11)| > 37149801960.55.

As far as we know, it has not been proved if e^^Pi is an irrational number (or not).

Examples

For n = 3, a(3) = floor(e[3]Pi) = floor(e^Pi) = 15.

Links

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung phi(phi(x)) = e^x und verwandter Funktionalgleichungen, J. reine angew. Math. 187, 56-67 (1950)

Sheldon Levenstein (user sheldonison), New fatou.gp program, Jul 10 2015, updated Aug 14 2019.

William Paulsen, Tetration.

William Paulsen, Tetration for complex bases, Advances in Computational Mathematics, Vol. 45, No. 1 (2019), pp. 243-267; ResearchGate link.

Wikipedia, Hyperoperation

Wikipedia, Tetration

Crossrefs

Cf. A000796, A001113, A019609, A039661, A059742, A351727, A353246.

Formula

a(n) = floor(e[n]Pi).

Showing 1-10 of 25 results. Next

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A073244 Decimal expansion of Pi - e.

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Examples

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Programs

Mathematica

PARI

A058651 Continued fraction for Pi + e.

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Keywords

Comments

Examples

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Crossrefs

Programs

PARI

PARI

Extensions

A094001 Decimal expansion of (Pi + e)^Pi.

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Programs

Mathematica

PARI

Sage

A121831 a(n) = floor((Pi + e)^n).

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Mathematica

Extensions

A121899 a(n) = ceiling((Pi + e)*sqrt(n)).

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Programs

Magma

Mathematica

PARI

Extensions

A121901 a(n) = floor((Pi + e)*sqrt(n)).

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Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A291854 Decimal expansion of the "Pi AND e" (Piande) constant.

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Comments

Examples

Crossrefs

Formula

A291858 Decimal expansion of the "Pi OR e" (Piore) constant.

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Comments

Examples

Crossrefs

Formula