A039661 -id:A039661 - cp's OEIS frontend

A114844 Sum of first n digits of Pi to digit-wise power of first n digits of e.

9, 10, 14, 15, 40, 43046761, 43046763, 44726379, 44726404, 44732965, 44733590, 44766358, 432186847, 432186848, 432193409, 432193652, 432193656, 432193683, 432226451, 432226515, 432273171, 432273172, 432273208, 432338744, 432340931

Offset: 1

Jonathan Vos Post, Feb 19 2006

The 331st digit of Pi and the 331st digit of e are both 0, so to generate any additional terms of the sequence beyond 330 terms one would have to define 0^0 to be either 0 or 1. - Harvey P. Dale, Aug 05 2014

			Since Pi =
3.1415926535897932384626433832795028841971693993751058209749445923078164062...
and e =
2.71828182845904523536028747135266249775724709369995957496696762772407663...
we have:
a(1) = 9 = 3^2.
a(2) = 10 = 3^2 + 1^7.
a(3) = 14 = 3^2 + 1^7 + 4^1.
a(4) = 15 = 3^2 + 1^7 + 4^1 + 1^8.
a(5) = 40 = 3^2 + 1^7 + 4^1 + 1^8 + 5^2.
a(6) = 43046761 = 3^2 + 1^7 + 4^1 + 1^8 + 5^2 + 9^8.

Harvey P. Dale, Table of n, a(n) for n = 1..300
Eric Weisstein's World of Mathematics, Pi Digits.
Eric Weisstein's World of Mathematics, e.

Cf. A000796, A001113, A039661, A059850, A114605.

With[{nn=30},Accumulate[RealDigits[Pi,10,nn][[1]]^RealDigits[E,10,nn] [[1]]]] (* Harvey P. Dale, Aug 05 2014 *)

a(n) = Sum_{i=1..n} A000796(i)^A001113(i).

A216707 Decimal expansion of e^(2*Pi).

Original entry on oeis.org

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

Offset: 3

John W. Nicholson, Sep 16 2012

Gelfond's constant squared.

The proportionate distance between images of an object (e.g., galaxy) seen behind a nonrotating (Schwarzschild) black hole, see Sneppen link. The factor is reduced for rotating (Kerr) black holes. - Charles R Greathouse IV, Oct 17 2021

			535.4916555247647365030493295890471814778057976032949155072052550373...

Albert Sneppen, Divergent reflections around the photon sphere of a black hole, Scientific Reports 11, 14247 (2021).
Index entries for transcendental numbers

Cf. A039661.

RealDigits[E^(2*Pi), 10, 105][[1]] (* T. D. Noe, Sep 17 2012 *)

exp(2*Pi) \\ Charles R Greathouse IV, May 14 2019

Equals A039661^2.

Equals (i^i)^(-4).

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

Marco Ripà, Apr 08 2022

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).

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

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

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

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

A094078 Decimal expansion of Pi + arctan(e^Pi).

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 30 2004

This is an approximation to the Feigenbaum constant.

			4.669201931859907731558852626...

Eric Weisstein's World of Mathematics, Feigenbaum constant.

Cf. A006890, A039661.

RealDigits[Pi + ArcTan[E^Pi], 10, 120][[1]] (* Robert G. Wilson v, May 04 2004 *)

More terms from Klaus Brockhaus and Robert G. Wilson v, May 03 2004

A104791 In decimal expansion of exp(Pi), positions of 10-digit partitions containing exactly 10 distinct digits.

Original entry on oeis.org

12186, 13451, 15422, 27621, 27622, 32790, 33051, 35249, 38760

Offset: 1

Zak Seidov, Mar 25 2005

			p[ 12186 ]= {8 4 2 7 0 3 5 6 9 1},
p[ 13451 ]= {0 7 9 5 3 4 1 6 2 8},
p[ 15422 ]= {0 2 9 1 3 8 7 5 6 4},
p[ 27621 ]= {9 3 2 5 8 4 1 0 7 6},
p[ 27622 ]= {3 2 5 8 4 1 0 7 6 9},
p[ 32790 ]= {2 4 0 8 6 5 3 7 1 9},
p[ 33051 ]= {6 2 5 3 9 0 4 1 8 7},
p[ 35249 ]= {6 3 7 2 5 8 9 4 0 1},
p[ 38760 ]= {3 1 0 6 9 2 5 4 7 8}.

Cf. A039661, A104781, A104790.

A277092 Decimal expansion of e^Pi/Pi^e.

Original entry on oeis.org

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

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

			1.030345524216210832441552437544142391331167453542635047752...

Cf. A000796, A001113, A039661, A059850.

RealDigits[N[E^Pi/Pi^E, 120]] (* Michael De Vlieger, Sep 30 2016 *)

More digits from Jon E. Schoenfield, Mar 15 2018

A372719 Decimal expansion of Sum_{k >= 1} floor(ke^(Pisqrt(163/9)))/2^k.

Original entry on oeis.org

1, 2, 8, 0, 6, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 7

Paolo Xausa, May 11 2024

nonn

			1280639.9999999999999999999999999999999999999999999999999999...

J. M. Borwein and P. B. Borwein, Strange Series and High Precision Fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.

Cf. A000796, A001113, A038152, A190575, A039661.

First[RealDigits[Sum[Floor[k*E^(Pi*Sqrt[163/9])]/2^k, {k, 400}], 10, 100]]

Approximately 1280640, correct to at least half a billion digits: see Sum 10 in Borwein and Borwein (1992), p. 623.

A380965 Decimal expansion of the solution to e^(x+Pi) = Pi^(x+e).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Feb 09 2025

It would be nice to have a reference to the scientific literature for this number. - N. J. A. Sloane, Feb 11 2025

			0.2065517167786496639130717563086969850230914439...

Vicente, "x" approximate value?. [Posting to X, formerly Twitter. Author's name? Date?]

Cf. A000796, A001113, A039661, A053510, A059850, A179701.

RealDigits[(Pi-E Log[Pi])/(Log[Pi]-1),10,100][[1]]

Equals (Pi - e*log(Pi))/(log(Pi) - 1).

cp's OEIS Frontend

A114844 Sum of first n digits of Pi to digit-wise power of first n digits of e.

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A216707 Decimal expansion of e^(2*Pi).

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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).

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A094078 Decimal expansion of Pi + arctan(e^Pi).

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Extensions

A104791 In decimal expansion of exp(Pi), positions of 10-digit partitions containing exactly 10 distinct digits.

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A277092 Decimal expansion of e^Pi/Pi^e.

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A372719 Decimal expansion of Sum_{k >= 1} floor(k*e^(Pi*sqrt(163/9)))/2^k.

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A380965 Decimal expansion of the solution to e^(x+Pi) = Pi^(x+e).

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A372719 Decimal expansion of Sum_{k >= 1} floor(ke^(Pisqrt(163/9)))/2^k.