A039661 -id:A039661 - cp's OEIS frontend

A334401 Decimal expansion of sinh(Pi).

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi - e^(-Pi))/2 = 11.5487393572577483779773343153884...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073742, A093580, A156648, A323983, A334402.

Mathematica
```
RealDigits[Sinh[Pi], 10, 110] [[1]]
```

Equals Sum_{k>=0} Pi^(2*k+1)/(2*k+1)!.

Equals 2 * Product_{k>=1} (4*k^2+4)/(4*k^2-1).

A068470 Decimal expansion of exp(sqrt(Pi)).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 10 2002

			5.8852772500180288766117618534057698039906986189859...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002161 (sqrt(Pi)), A039661 (exp(Pi)).

SetDefaultRealField(RealField(100)); R:=RealField(); Exp(Sqrt(Pi(R))); // G. C. Greubel, Nov 27 2018

evalf[120](exp(sqrt(Pi))); # Muniru A Asiru, Nov 28 2018

RealDigits[Exp[Sqrt[Pi]],10,120][[1]] (* Harvey P. Dale, Aug 22 2012 *)

default(realprecision, 100); exp(sqrt(Pi)) \\ G. C. Greubel, Jan 12 2017

numerical_approx(exp(sqrt(pi)), digits=100) # G. C. Greubel, Nov 27 2018

A166741 E.g.f.: exp(2*arcsin(x)).

Original entry on oeis.org

1, 2, 4, 10, 32, 130, 640, 3770, 25600, 199810, 1740800, 16983850, 181043200, 2122981250, 26794393600, 367275756250, 5358878720000, 84106148181250, 1393308467200000, 24643101417106250, 457005177241600000

Offset: 0

Jaume Oliver Lafont, Oct 21 2009

nonn

exp(2*arcsin(1)) is Aleksandr Gelfond's constant.

Wikipedia, Gelfond's constant

Cf. A006228, A039661, A166748.

seq(simplify(2^(n-1) * (cosh(Pi)*(1-(-1)^n) + sinh(Pi)*(1+(-1)^n)) * GAMMA((1/2)*n-I)*GAMMA((1/2)*n+I) / Pi), n=0..20); # Vaclav Kotesovec, Nov 06 2014

CoefficientList[Series[E^(2*ArcSin[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
FullSimplify[Table[2^(n-1) * (E^(Pi)-(-1)^n*E^(-Pi)) * Gamma[n/2-I] * Gamma[n/2+I] / Pi,{n,0,20}]] (* Vaclav Kotesovec, Nov 06 2014 *)

for (n=0,25,print(polcoeff(exp(2*asin(x)),n)*n!,","))

a(n) ~ 2 * n^(n-1) * (exp(Pi) - (-1)^n/exp(Pi)) / exp(n). - Vaclav Kotesovec, Aug 04 2014
From Vaclav Kotesovec, Nov 06 2014: (Start)
a(n) = (n^2 - 4*n + 8)*a(n-2).
a(n) = 2^(n-1) * (exp(Pi)-(-1)^n*exp(-Pi)) * GAMMA(n/2-I) * GAMMA(n/2+I) / Pi.
(End)

A334402 Decimal expansion of cosh(Pi).

Original entry on oeis.org

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi + e^(-Pi))/2 = 11.5919532755215206277517520525601...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073743, A093580, A175314, A308715, A323982, A334401.

Mathematica
```
RealDigits[Cosh[Pi], 10, 110] [[1]]
```

Equals Sum_{k>=0} Pi^(2*k)/(2*k)!.
Equals Product_{k>=0} (1 + 4/(2*k+1)^2).
Equals Product_{k>=1} (k^2 + 4)/(k^2 + 1). - Amiram Eldar, Aug 09 2020

A038152 Beatty sequence for e^Pi.

Original entry on oeis.org

23, 46, 69, 92, 115, 138, 161, 185, 208, 231, 254, 277, 300, 323, 347, 370, 393, 416, 439, 462, 485, 509, 532, 555, 578, 601, 624, 647, 671, 694, 717, 740, 763, 786, 809, 833, 856, 879, 902, 925, 948, 971, 995, 1018, 1041, 1064

Offset: 1

Felice Russo

Cf. A000796, A001113, A039661.

A038152:=n->floor(n*exp(Pi)): seq(A038152(n), n=1..100); # Wesley Ivan Hurt, Jan 21 2017

Floor[Range[100]*E^Pi] (* Paolo Xausa, May 11 2024 *)

a(n) = floor(n*exp(Pi)); \\ Michel Marcus, Sep 28 2013

a(n) = floor(n*e^Pi).

Corrected and extended by Carlos Alves, Nov 25 2006

A104691 Decimal expansions of e and Pi interlaced.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Apr 23 2005

Within 2.5 per cent of e^Pi (A039661: 23.14069...) or 5.5 per cent of Pi^e (A059850: 22.45915...). - Robert G. Wilson v, Jan 04 2013

Cf. A001355.

Riffle[RealDigits[E, 10, 53][[1]], RealDigits[Pi, 10, 53][[1]]] (* Robert G. Wilson v, Jan 04 2013 *)

A105007 Primes from merging of 2 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

23, 31, 79, 29, 67, 79, 47, 73, 61, 11, 19, 43, 23, 83, 97, 71, 19, 97, 67, 19, 67, 59, 73, 31, 41, 83, 47, 71, 17, 79, 53, 79, 23, 89, 13, 37, 41, 17, 41, 83, 47, 43, 59, 43, 67, 41, 13, 67, 71, 31, 19, 41, 47, 11, 37, 73, 31, 47, 47, 73, 53, 83, 31, 29, 47, 89, 19, 43, 73

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 31 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
The first 5,000 digits of exp(PI) as calculated by _Simon Plouffe_ at WorldWideSchool.org.
Eric Weisstein, Exponential Functions

Cf. A039661.

Select[FromDigits /@ Partition[RealDigits[Exp[Pi], 10, 500][[1]], 2, 1], # > 9 && PrimeQ[#] &] (* Vincenzo Librandi, Apr 26 2013 *)

Changed offset from 0 to 1 by Vincenzo Librandi, Apr 26 2013

A105014 Primes from merging of 9 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

269005729, 290863679, 473802661, 527835169, 706754921, 754921967, 801087773, 835844717, 717445879, 796098493, 849365327, 327965863, 991013741, 984449143, 143096677, 136716319, 128763773, 377314703, 833162821, 294047891, 654433627

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 31 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
The first 5,000 digits of exp(PI) as calculated by _Simon Plouffe_ at WorldWideSchool.org.
Eric Weisstein, Exponential Functions

Cf. A039661.

Select[FromDigits /@ Partition[RealDigits[Exp[Pi], 10, 500][[1]], 9, 1], # > 99999999 && PrimeQ[#]&] (* Vincenzo Librandi, Apr 27 2013 *)

Changed offset from 0 to 1 by Vincenzo Librandi, Apr 27 2013

A194554 Decimal expansion of the absolute value of the imaginary part of i^(e^Pi), where i = sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Aug 28 2011

If Schanuel's Conjecture is true, then i^e^Pi is transcendental (see Marques and Sondow 2010, p. 79).

			i^e^Pi = 0.2192048949... - 0.9756788478...*i

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 1.1
D. Marques and Jonathan Sondow, Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, https://arxiv.org/abs/1010.6216 [math.NT], 2010-2011; East-West J. Math., 12 (2010), 75-84.
Wikipedia, Schanuel's conjecture

Cf. A039661 (decimal expansion of e^Pi), A194555 (real part).

RealDigits[Im[I^E^Pi], 10, 100] // First

abs(imag(I^(exp(Pi)))) \\ Michel Marcus, Aug 19 2018

A225142 Primes from merging of 10 successive digits in the decimal expansion of exp(Pi).

Original entry on oeis.org

1406926327, 2632779269, 9269005729, 9045278351, 9706754921, 2196759527, 3835844717, 6468440141, 6844014103, 8216511287, 8763773147, 3473538331, 6273655727, 9618858059, 5430487121, 1300849327, 9325853203, 6189528329, 6802254197, 3054479927, 4479927817

Offset: 1

Bruno Berselli, Apr 30 2013

Leading zeros are not permitted, so each prime is 10 digits in length. The terms are listed in the order in which they occur.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Simon Plouffe, exp(Pi) to 5000 digits

Cf. A039661, A105007 - A105014.

Cf. A105015.

With[{len = 10}, FromDigits /@ Select[Partition[RealDigits[E^Pi, 10, 600][[1]], len, 1], PrimeQ[FromDigits[#]] && IntegerLength[FromDigits[#]] == len &]]
Select[FromDigits/@Partition[RealDigits[Exp[Pi],10,600][[1]],10,1],PrimeQ[ #] && IntegerLength[#]==10&] (* Harvey P. Dale, Aug 27 2020 *)

cp's OEIS Frontend

A334401 Decimal expansion of sinh(Pi).

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A068470 Decimal expansion of exp(sqrt(Pi)).

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Magma

Maple

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PARI

Sage

A166741 E.g.f.: exp(2*arcsin(x)).

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Maple

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PARI

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A334402 Decimal expansion of cosh(Pi).

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A038152 Beatty sequence for e^Pi.

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Maple

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PARI

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Extensions

A104691 Decimal expansions of e and Pi interlaced.

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Mathematica

A105007 Primes from merging of 2 successive digits in decimal expansion of exp(Pi).

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Mathematica

Extensions

A105014 Primes from merging of 9 successive digits in decimal expansion of exp(Pi).

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