A072334 -id:A072334 - cp's OEIS frontend

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

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

A278327 Decimal expansion of 1/e - 1/e^2.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Nov 18 2016

The probability, as n = 2^k increases without bound, that a randomized skip list with n elements and p = 1/2 has exactly k levels.

			0.232544157934829629701524275188976464...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Skip list.

Cf. A001113, A068985, A072334, A091131, A092553, A158466.

RealDigits[1/E - 1/E^2, 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)

Equals 1/A001113 - 1/A072334 = A068985 - A092553.

Equals A091131 / A072334 = A091131 * A092553.

A367574 Decimal expansion of BesselI(0,2*sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Nov 28 2023

			4.252350879502623825293230824089510302...

Cf. A000217, A006472, A070910, A072334, A367709, A367710.

RealDigits[BesselI[0, 2 Sqrt[2]], 10, 100][[1]]

Equals Sum_{k>=0} 2^k / k!^2.

Equals Sum_{i>=0} (i+1)/Product_{j=1..i} A000217(j). - Davide Rotondo, Feb 25 2025

A104999 Primes from merging of 3 successive digits in decimal expansion of exp(2).

Original entry on oeis.org

389, 227, 131, 557, 127, 257, 379, 607, 577, 431, 179, 947, 773, 547, 661, 127, 733, 337, 839, 607, 107, 239, 947, 269, 647, 523, 487, 757, 541, 467, 281, 293, 331, 101, 193, 337, 997, 953, 307, 751, 823, 947, 479, 991, 587, 877, 683, 239, 727, 883, 461

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(2) as calculated by _Simon Plouffe_ at WorldWideSchool.org. [broken link]
Eric Weisstein, Exponential Functions

Cf. A072334

 Select[FromDigits/@Partition[RealDigits[Exp[2], 10, 500][[1]], 3, 1],#>99&&PrimeQ[#]&] (* Vincenzo Librandi, Apr 26 2013 *)

A105003 Primes from merging of 7 successive digits in decimal expansion of exp(2).

Original entry on oeis.org

7500781, 8031557, 1217947, 2179477, 1794773, 4788661, 7812733, 3913309, 6254141, 9928129, 3188807, 3301019, 3378997, 8997407, 9740729, 7407299, 9600953, 1532081, 2368469, 6846947, 4793029, 4456831, 6831239, 9996461

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(2) as calculated by _Simon Plouffe_ at WorldWideSchool.org. [broken link]
Eric Weisstein, Exponential Functions

Cf. A072334

 Select[FromDigits/@Partition[RealDigits[Exp[2], 10, 600][[1]], 7, 1], #>999999&&PrimeQ[#]&] (* Vincenzo Librandi, Apr 26 2013 *)

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

A105006 Erroneous version of: Primes from merging of 10 successive digits in decimal expansion of exp(2).

Original entry on oeis.org

1784090467, 1183898827, 1445683123, 1370213483, 1571224183, 1193531621, 2117195173, 1316084201, 1109265013, 1849899449, 1609339583, 1862200777, 1211998849, 1071519781, 1654739179, 2006107949, 2050796009, 1068857957

Offset: 1

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

See A225141 for the correct one. What is this sequence? [Bruno Berselli, Apr 30 2013]

The first 5,000 digits of exp(2) as calculated by _Simon Plouffe_ at WorldWideSchool.org. [broken link]
Eric Weisstein, Exponential Functions

Cf. A072334

A234604 Floor of the solutions to c = exp(1 + n/c) for n >= 0, using recursion.

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 6, 7, 17, 35, 62, 103, 164, 256, 391, 589, 880, 1303, 1919, 2814, 4112, 5993, 8716, 12655, 18353, 26591, 38499, 55710, 80583, 116523, 168453, 243485, 351889, 508506, 734776, 1061672, 1533938, 2216216

Offset: 0

Richard R. Forberg, Dec 28 2013

nonn

For n = 1 to 7 recursion produces convergence to single valued solutions.
For n >= 8 a dual-valued oscillating recursion persists between two stable values. The floor of the upper value for each n is included here. (The lower values of c are under 6 and approach exp(1) = 2.71828 for large n.)
At large n, the ratio of a(n)/a(n-1) approaches exp(1/exp(1)) = 1.444667861009 with more digits given by A073229.
At n = 0, c = exp(1).
At n = 1, c = 3.5911214766686 = A141251.
At n = 2, c = 4.3191365662914
At n = 3, c = 4.9706257595442
At n = 4, c = 5.5723925978776
At n = 5, c = 6.1383336446072
At n = 6, c = 6.6767832796664
At n = 7, c = 7.1932188286406
The convergence becomes "dual-valued" at n > exp(2) = 7.3890560989 = A072334.
At values of n = 7 and 8 the convergence is noticeably slower than at either larger or smaller values of n.
The recursion at n = exp(2) is only "quasi-stable" where c reluctantly approaches exp(2) = exp(1 + exp(2)/exp(2)) from any starting value, but never reaches it, and is not quite able to hold it if given the solution, due to machine rounding errors.

Cf. A073229, A141251.

a(n) = floor(c) for the solutions to c = exp(1 + n/c) at n = 0 to 7, and the floor of the stable upper values of c for n >= 8.

Conjecture: a(n) = floor(e^(-e^(t^2/e^t - t)*t^2 + t + 1)) for all n > 13. - Jon E. Schoenfield, Jan 11 2014

Corrected and edited by Jon E. Schoenfield, Jan 11 2014

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Jul 10 2021

			0.2288989481961786412366361253722...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A072334, A073742, A073747.

SetDefaultRealField(RealField(135)); 2/(Exp(2)-1)*Exp(2/(1-Exp(2))); // G. C. Greubel, Jun 11 2024

x/.FindRoot[LambertW[-x]-LambertW[-1,-x]==2, {x, 0.1, 0.3}, WorkingPrecision -> 110]
RealDigits[2/(E^2-1)*Exp[2/(1-E^2)], 10, 135][[1]] (* G. C. Greubel, Jun 11 2024 *)

PARI
```
exp(-cotanh(1))/sinh(1)
```

numerical_approx(2/(e^2-1)*exp(2/(1-e^2)), digits=135) # G. C. Greubel, Jun 11 2024

Equals exp(-coth(1))/sinh(1) = exp(-A073747)/A073742.
Equals (coth(1)-1)*exp(1-coth(1)) = (A073747-1)*exp(1-A073747).
Equals (coth(1)+1)/exp(1+coth(1)) = (A073747+1)/exp(1+A073747).
Equals 2/(e^2-1)*exp(2/(1-e^2)) = 2/(A072334^2-1)*exp(2/(1-A072334^2)).

A353246 Integer part of e[n]e, where [n] indicates hyper-n and e = 2.718281828... (using H. Kneser's proposal for n > 3).

Original entry on oeis.org

4, 5, 7, 15, 2075

Offset: 0

Marco Ripà, Apr 08 2022

The common hyperoperation sequence is defined as follows: hyper-0 = zeration, hyper-1 = addition, hyper-2 = multiplication, hyper-3 = exponentiation, hyper-4 = tetration, and so on...
Thus e[0]e = e + 2, e[1]e = 2*e, e[2]e = e^2, e[3]e = e^e, and so on.
The fifth term of the twin sequence of the present one, floor(Pi[4]Pi), is much larger than 2075 and it is harder to calculate, while the integer part of e[4]Pi is 37149801960 (17.9 million times bigger than a(4)).

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

Cf. A001113, A019762, A072334, A073226, A351727, A352396.

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

A096438 Decimal expansion of (Pi^2 - e^2)^(1/3).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Aug 10 2004

			1.3536796372073169512106569366...

Cf. A002388, A072334, A096255.

cp's OEIS Frontend

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

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A278327 Decimal expansion of 1/e - 1/e^2.

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A367574 Decimal expansion of BesselI(0,2*sqrt(2)).

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A104999 Primes from merging of 3 successive digits in decimal expansion of exp(2).

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A105003 Primes from merging of 7 successive digits in decimal expansion of exp(2).

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A105006 Erroneous version of: Primes from merging of 10 successive digits in decimal expansion of exp(2).

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A234604 Floor of the solutions to c = exp(1 + n/c) for n >= 0, using recursion.

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Extensions

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

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A353246 Integer part of e[n]e, where [n] indicates hyper-n and e = 2.718281828... (using H. Kneser's proposal for n > 3).

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