A039661 -id:A039661 - cp's OEIS frontend

A019315 Decimal expansion of e^Pi + Pi + e.

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

Offset: 2

Dan Hoey

			29.00056711482810747955201722258071276222052273...

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A000796, A001113, A039661.

First[RealDigits[E^Pi + Pi + E, 10, 100]] (* Paolo Xausa, May 02 2024 *)

A059196 Engel expansion of e^Pi = 23.14069... .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 232, 238, 428, 1103, 3261, 6085, 15565, 20674, 43910, 177426, 193017, 1480418, 1739984, 2573921, 5238757, 14403086, 25953812, 34670065, 84077823, 258624998, 330484686

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1023
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A039661.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[E^Pi, 7!], 100] (* G. C. Greubel, Dec 28 2016 *)

A104790 Lengths of the sections of decimal expansion of exp(Pi) containing all 10 digits at least once.

Original entry on oeis.org

25, 17, 40, 29, 26, 23, 22, 31, 30, 27, 31, 28, 31, 20, 13, 24, 17, 30, 37, 39, 30, 24, 28, 40, 20, 44, 32, 21, 18, 19, 18, 65, 26, 20, 24, 51, 25, 35, 37, 23, 39, 23, 20, 32, 25, 28, 16, 33, 20, 34, 44, 41, 32, 21, 31, 43, 17, 33, 17, 30, 44, 35, 36, 45, 19, 13, 36, 24, 17

Offset: 1

Zak Seidov, Mar 25 2005

Among first 500 sections, s483 has the minimal length=12. In general, in decimal expansion of exp(pi), the first three partitions of 10 distinct digits, are: 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}.

			s1={2,3,1,4,0,6,9,2,6,3,2,7,7,9,2,6,9,0,0,5,7,2,9,0,8}-25 digits,
s2={6,3,6,7,9,4,8,5,4,7,3,8,0,2,6,6,1}-17 digits,
s3={2,3,1,4,0,6,9,2,6,3,2,7,7,9,2,6,9,0,0,5,7,2,9,0,8}-40 digits,
s483={4,7,8,1,3,5,6,6,2,4,0,9}- 12 digits.

Cf. A039661, A104781.

A105008 Primes from merging of 3 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

263, 277, 269, 863, 367, 547, 661, 211, 199, 409, 971, 719, 199, 997, 967, 877, 773, 383, 587, 653, 863, 223, 991, 101, 137, 401, 103, 881, 449, 449, 491, 677, 367, 163, 631, 821, 773, 347, 353, 383, 331, 821, 919, 193, 443, 433, 557, 727, 271, 823, 449

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]], 3, 1], #>99&&PrimeQ[#]&] (* Vincenzo Librandi, Apr 26 2013 *)

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

A105009 Primes from merging of 4 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

6367, 1993, 1697, 6971, 9719, 1997, 7549, 4801, 1087, 4441, 8447, 5879, 2689, 6899, 1013, 7417, 1039, 7243, 8059, 5881, 7841, 1367, 3671, 3821, 4703, 3833, 2129, 4789, 5443, 1823, 8237, 8059, 4547, 5479, 7901, 9013, 2659, 5101, 1499, 9643, 4229, 1607

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]], 4, 1], # > 999 && PrimeQ[#] &] (* Vincenzo Librandi, Apr 27 2013 *)

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

A105010 Primes from merging of 5 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

69263, 63277, 32779, 90863, 35069, 52783, 77731, 69383, 44587, 79609, 65327, 53279, 24223, 84401, 41039, 68477, 77243, 68059, 95881, 84449, 44491, 16319, 31963, 34147, 11287, 63773, 47353, 82129, 78919, 57271, 13397, 65951, 95101

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]], 5, 1], # > 9999 && PrimeQ[#] &] (* Vincenzo Librandi, Apr 27 2013 *)

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

A105011 Primes from merging of 6 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

926327, 948547, 802661, 235069, 754921, 704801, 358447, 584471, 844717, 365327, 103951, 309667, 367163, 478403, 821651, 773147, 470347, 353833, 538331, 162821, 433627, 655727, 498961, 547901, 901339, 229681, 681607, 565381, 362069, 430487

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]], 6, 1], # > 99999 && PrimeQ[#] &] (* Vincenzo Librandi, Apr 27 2013 *)

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

A105012 Primes from merging of 7 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

9005729, 7290863, 5046409, 5243423, 3516971, 6759527, 7965863, 6924223, 1013741, 3868477, 8477243, 1128763, 8763773, 4789193, 3655727, 9168487, 2645479, 5479013, 7924229, 7995653, 5381423, 8142353, 9580729, 1691939

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]], 7, 1], # > 999999 && PrimeQ[#] &] (* Vincenzo Librandi, Apr 27 2013 *)

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

A105013 Primes from merging of 8 successive digits in decimal expansion of exp(Pi).

Original entry on oeis.org

29086367, 67948547, 79485473, 45278351, 67549219, 41469383, 71744587, 27965863, 68477243, 44984449, 49844491, 11287637, 63773147, 77314703, 70347353, 29404789, 18237449, 23744989, 72645479, 83402659, 65381423

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]], 8, 1], # > 9999999 && PrimeQ[#] &] (* Vincenzo Librandi, Apr 27 2013 *)

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

A114605 Sum of first n digits of e to digit-wise power of first n digits of Pi.

Original entry on oeis.org

8, 15, 16, 24, 56, 134217784, 134217785, 134479929, 134479961, 134480473, 134481497, 134872122, 522292611, 522292611, 522554755, 522554880, 522554884, 522554911, 522945536, 522945617

Offset: 1

Jonathan Vos Post, Feb 17 2006

			Since e = 2.71828182845904523536028747135266249775724709369995957496696762772407663...
and Pi =
3.1415926535897932384626433832795028841971693993751058209749445923078164062...
a(1) = 8 = 2^3.
a(2) = 15 = 2^3 + 7^1.
a(3) = 16 = 2^3 + 7^1 + 1^4.
a(4) = 24 = 2^3 + 7^1 + 1^4 + 8^1.
a(5) = 56 = 2^3 + 7^1 + 1^4 + 8^1 + 2^5.
a(6) = 134217784 = 2^3 + 7^1 + 1^4 + 8^1 + 2^5 + 8^9.

Eric Weisstein's World of Mathematics, Pi Digits.
Eric Weisstein's World of Mathematics, e.

Cf. A000796, A001113, A039661.

Module[{nn=20,pi,ee},pi=RealDigits[Pi,10,nn][[1]];ee=RealDigits[E, 10, nn] [[1]]; Table[Total[Take[ee,n]^Take[pi,n]],{n,nn}]] (* Harvey P. Dale, Jan 13 2017 *)

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

cp's OEIS Frontend

A019315 Decimal expansion of e^Pi + Pi + e.

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A059196 Engel expansion of e^Pi = 23.14069... .

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A104790 Lengths of the sections of decimal expansion of exp(Pi) containing all 10 digits at least once.

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A105008 Primes from merging of 3 successive digits in decimal expansion of exp(Pi).

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A105009 Primes from merging of 4 successive digits in decimal expansion of exp(Pi).

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A105010 Primes from merging of 5 successive digits in decimal expansion of exp(Pi).

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A105011 Primes from merging of 6 successive digits in decimal expansion of exp(Pi).

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A105012 Primes from merging of 7 successive digits in decimal expansion of exp(Pi).

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A105013 Primes from merging of 8 successive digits in decimal expansion of exp(Pi).

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