A202955 -id:A202955 - cp's OEIS frontend

A085667 Decimal expansion of e^e^e^e.

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

Offset: 1656521

N. J. A. Sloane, Jul 15 2003

			2.331504399007195462289689911012137666332017428963... * 10^1656520

J. Blanck, Exact real arithmetic systems: results of competition, pp. 389-393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science, Springer-Verlag, 2001.
Hans Havermann, integer part of e^e^e^e (in blocks of 100 digits) = the first 1656521 terms
Hans Havermann, fractional part of e^e^e^e (in blocks of 100 digits) = an additional 1656521 terms

Cf. A073226, A073227, A073228, A181180, A073232, A202955.

RealDigits[Exp[Exp[Exp[Exp[1]]]],10,120][[1]] (* Harvey P. Dale, Sep 06 2012 *)

exp(exp(exp(exp(1)))) \\ Charles R Greathouse IV, Mar 25 2014

A241293 Decimal expansion of 4^(4^(4^4)) = 4^^4.

Original entry on oeis.org

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

Offset: 1

Robert Munafo and Robert G. Wilson v, Apr 18 2014

The offset is 1 because the true offset would be 8.072304726...*10^153, which is too large to be represented properly in the OEIS.

This is the decimal expansion of 2^2^513. - Jianing Song, Dec 25 2018

			2361022671459731320687702749778179430946127291477515446719256946212711853666475524945769350101941997...(8.072304726...*10^153) ... 7470426497333490366540651560537534642789067586985427238232605843019607448189676936860456095261392896.
The above line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parenthesis.
The final one hundred digits where computed by: PowerMod[4, 4^4^4, 10^100].

Robert P. Munafo, Hyper4 Iterated Exponential Function..

Cf. A114561, A085667, A202955, A054382, A014221, A241291, A241292, A241294, A241295, A241296, A241297, A241298, A241299, A243913.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[ 4, 4^4^4] (* or *)
p = 4; f[n_] := Quotient[n^p, 10^(Floor[p * Log10@ n] - (1004 + p^p))]; IntegerDigits@ Quotient[ Nest[ f@ # &, p, p^p], 10^(900 + p^p)] (* Program fixed by Jianing Song, Sep 18 2019 *)

4^(4^(4^4)) = ((((( ... 245 ... (((((4^4)^4)^4)^4)^4) ... 245 ... ^4)^4)^4)^4)^4)^4.

Keyword: fini added by Jianing Song, Sep 18 2019

A241291 Decimal expansion of 2^(2^(2^(2^(2^2)))) = 2^^6.

Original entry on oeis.org

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

Offset: 1

Robert Munafo and Robert G. Wilson v, Apr 18 2014

The offset is 1 because the true offset would be 6.0312260626165015 * 10^19727, which is too large to be represented properly in the OEIS.
2^0 = 1, 2^1 = 2, 2^2 = 4,
2^2^2 = 2^^3 = (2^2)^2 = 16,
2^2^2^2 = 2^^4 = (((2^2)^2)^2)^2 = 65536,
2^(2^(2^(2^2))) = 2^^5 = (((((((((((((((2^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2 =
2003529930406846464979072351560255750447825475569751419265016973710894059556311453089506130880933348...(19529 digits)...9087575630505718260979581044520267611188489786293085833548068862693010305614986891826277507437428736.

			2120038728808211984885164691662274630835654230675372483625951752354414565561161040708771008806932213...(10^(6.0312260626165015 * 10^19727))...9087575630505718260979581044520267611188489786293085833548068862693010305614986891826277507437428736.
The above example line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parentheses.

Robert P. Munafo, Sequence A094358, 2^^N = 1 mod N.
Robert P. Munafo, Hyper4 Iterated Exponential Function.

Cf. A014221, A085667, A202955, A054382, A014221, A241292, A241293, A241294, A241295, A241296, A241297, A241298, A241299, A243913.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[2, 2^2^2^2^2]

Equals 2^2^2^2^2^2 = 2^^6.

Keyword: fini added by Jianing Song, Sep 18 2019

A241292 Decimal expansion of 3^(3^(3^3)) = 3^^4.

Original entry on oeis.org

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

Offset: 3638334640025

Robert Munafo and Robert G. Wilson v, Apr 18 2014

Decimal expansion of 3^7625597484987. - Jianing Song, Sep 15 2019

			=1258014290627491317860390698203281215518046714316596015189674944381211011300017785310803903296240115...(3638334639825)...5344828628021555146929939999502212249640012905650177570718344711077047886315075206738945776100739387.
The above example line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parenthesis.
The final one hundred digits where computed by: PowerMod[3, 3^3^3, 10^100].

Robert P. Munafo, Hyper4 Iterated Exponential Function..

Cf. A000244, A014222.
Cf. A085667, A202955,
Cf. A054382, A014221, A241291, A241293, A241294, A241295, A241296, A241297, A241298, A241299.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[ 3, 3^3^3] (* or *)
p = 3; f[n_] := Quotient[n^p, 10^(Floor[p * Log10@ n] - (1004 + p^p))]; IntegerDigits@ Quotient[ Nest[ f@ # &, p, p^p], 10^(900 + p^p)]

3.^3^3^3 \\ Charles R Greathouse IV, Apr 25 2016

= 3^(3^(3^3)) = ((((( ... 16 ... (((((3^3)^3)^3)^3)^3) ... 16 ... ^3)^3)^3)^3)^3)^3.

Keyword: fini added by Jianing Song, Sep 18 2019

A241294 Decimal expansion of 5^(5^(5^5)) = 5^^4.

Original entry on oeis.org

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

Offset: 1

Robert Munafo and Robert G. Wilson v, Apr 18 2014

The offset is 1 because the true offset would be 1.335740484... * 10^2184, which is too large to be represented properly in the OEIS.

			1111028808179997445286178274186057545167346520596272154733386745225196554833740184735209940181105736...(1.335740484... * 10^2184)...3293393812245587348839009777160541868907233602002347435809721798438687301313620992004871368408203125.
The above line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parenthesis.
The final one hundred digits where computed by: PowerMod[5, 5^5^5, 10^100].

Robert P. Munafo, Hyper4 Iterated Exponential Function..

Cf. A085667, A202955, A054382, A014221, A241291, A241292, A241293, A241295, A241296, A241297, A241298, A241299, A243913.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[ 5, 5^5^5] (* or *)
p = 5; f[n_] := Quotient[n^p, 10^(Floor[p * Log10@ n] - (1004 + p^p))]; IntegerDigits@ Quotient[ Nest[ f@ # &, p, p^p], 10^(900 + p^p)]

5^(5^(5^5)) = ((((( ... 3114 ... (((((5^5)^5)^5)^5)^5) ... 3114 ... ^5)^5)^5)^5)^5)^5.

Keyword: fini added by Jianing Song, Sep 18 2019

A241295 Decimal expansion of 6^(6^(6^6)) = 6^^4.

Original entry on oeis.org

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

Offset: 1

Robert Munafo and Robert G. Wilson v, Apr 18 2014

The offset is 1 because the true offset would be 2.069197376... * 10^36305, which is too large to be represented properly in the OEIS.

			4446235190236934697525658222829200154057941454246396634278261569446314669723229775849294305048219193...(2.069197376... * 10^36305)...1753131067593004473155552781300975310520790674421755277191077815819279193580406457883859420138438656.
The above line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parenthesis.
The final one hundred digits where computed by: PowerMod[6, 6^6^6, 10^100].

Robert P. Munafo, Hyper4 Iterated Exponential Function..

Cf. A085667, A202955, A054382, A014221, A241291, A241292, A241293, A241294, A241296, A241297, A241298, A241299, A243913.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[ 6, 6^6^6] (* Program fixed by Jianing Song, Sep 18 2019 *)

6^(6^(6^6)) = ((((( ... 46645 ... (((((6^6)^6)^6)^6)^6) ... 46645 ... ^6)^6)^6)^6)^6)^6.

Keyword: fini added by Jianing Song, Sep 18 2019

A241296 Decimal expansion of 7^(7^(7^7)) = 7^^4.

Original entry on oeis.org

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

Offset: 1

Robert Munafo and Robert G. Wilson v, Apr 18 2014

The offset is 1 because the true offset would be 3.177419493... * 10^695974, which is too large to be represented properly in the OEIS.

			7833005237480055565403854094754653082919044398558770131482119703185028436339726344442972338289410045...(3.177419493... * 10^695974)...1766659134033863120639301828875443004447501140571853058472378511366058254036038182879357182733172343.
The above line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parenthesis.
The final one hundred digits where computed by: PowerMod[7, 7^7^7, 10^100].

Robert P. Munafo, Hyper4 Iterated Exponential Function..

Cf. A085667, A202955, A054382, A014221, A241291, A241292, A241293, A241294, A241295, A241297, A241298, A241299, A243913.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[ 7, 7^7^7]

7^(7^(7^7)) = ((((( ... 823532 ... (((((7^7)^7)^7)^7)^7) ... 823532 ... ^7)^7)^7)^7)^7)^7.

Keyword: fini added by Jianing Song, Sep 18 2019

A241297 Decimal expansion of 8^(8^(8^8)) = 8^^4.

Original entry on oeis.org

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

Offset: 1

Robert Munafo and Robert G. Wilson v, Apr 18 2014

The offset is 1 because the true offset would be 5.431653469... * 10^15151335, which is too large to be represented properly in the OEIS.

Decimal expansion of 2^(3*2^50331648). - Jianing Song, Dec 26 2022

			=6474032964669706799738662517939027493552465781556605471681845356387490969947645130386969932823714021...(5.431653456330093... * 10^15151335)...6641744766927456476257727570637041060682921214560194830819153337200429887920249826536946437619449856.
The above example line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parenthesis.
The final one hundred digits where computed by: PowerMod[8, 8^8^8, 10^100].

Robert P. Munafo, Hyper4 Iterated Exponential Function.

Cf. A085667, A202955, A054382, A014221, A241291, A241292, A241293, A241294, A241295, A241296, A241298, A241299, A243913.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[ 8, 8^8^8]

= 8^(8^(8^8)) = ((((( ... 16777205 ... (((((8^8)^8)^8)^8)^8) ... 16777205 ... ^8)^8)^8)^8)^8)^8.

Keyword: fini added by Jianing Song, Sep 18 2019

A241298 Decimal expansion of 9^(9^9) = 9^^3.

Original entry on oeis.org

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

Offset: 369693100

Robert Munafo and Robert G. Wilson v, Apr 18 2014

Decimal expansion of 3^774840978. - Jianing Song, Sep 15 2019

			= 42812477317574704803698711593056352133905548224144
  35141747537230535238874717350483531936652994320333
  ... (369,692,900 digits omitted) ...
  26170043150602250406601961656994397543610268552663
  74036682906190174923494324178799359681422627177289.
The first and last 100 digits are shown above, with the intervening digits omitted.
The final one hundred digits were computed using PowerMod[9, 9^9, 10^100].

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

Googlology Wiki, First Digits
Hans Havermann, n^n^n for n=1 to n=5
Hans Havermann, 9^9^9, Volume 1, the first x decimal digits, in 1100 pages with 10000 digits per page.
Math Forum, "Ask Dr. Math", Value of 9^(9^9)?
Robert P. Munafo, Hyper4 Iterated Exponential Function
Robert P. Munafo and Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi
Eric Weisstein's World of Mathematics, Joyce Sequence
Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi

Cf. A002488, A054382, A085667, A202955, A054382, A014221, A241291, A241292, A241293, A241294, A241295, A241296, A241297, A241299, A243913.

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[9, 9^9] (* or *)
f[n_] := Quotient[n^9, 10^(Floor[9*Log10@ n] - 1010)]; Nest[ f@ # &, 9, 9]

9^(9^9) = ((((((((9^9)^9)^9)^9)^9)^9)^9)^9)^9.

Keyword: fini added by Jianing Song, Sep 18 2019

A073234 Decimal expansion of Pi^(Pi^Pi).

Original entry on oeis.org

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

Offset: 19

Rick L. Shepherd, Jul 25 2002

			1340164183006357435.29744912964...

Christopher Creutzig & Walter Oevel, MuPAD Tutorial. Berlin: Springer-Verlag (2004): 339.

Harry J. Smith, Table of n, a(n) for n = 19..20000

Cf. A000796 (Pi), A073233 (Pi^Pi), A202955 (Pi^Pi^Pi^Pi), A073236 (Pi^Pi^...^Pi, n times, rounded), A073235 ((Pi^Pi)^Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)).

RealDigits[Pi^Pi^Pi, 10, 100][[1]] (* Alonso del Arte, Jul 03 2012 *)

DIGITS := 100:
float(PI^(PI^PI)) // from Creutzig & Oevel (2004)

PARI
```
Pi^Pi^Pi
```

{ default(realprecision, 20100); x=Pi^Pi^Pi/10^18; for (n=19, 20000, d=floor(x); x=(x-d)*10; write("b073234.txt", n, " ", d)); } \\ Harry J. Smith, May 01 2009, corrected May 19 2009, had -n

cp's OEIS Frontend

A085667 Decimal expansion of e^e^e^e.

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A241293 Decimal expansion of 4^(4^(4^4)) = 4^^4.

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A241291 Decimal expansion of 2^(2^(2^(2^(2^2)))) = 2^^6.

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A241292 Decimal expansion of 3^(3^(3^3)) = 3^^4.

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A241294 Decimal expansion of 5^(5^(5^5)) = 5^^4.

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A241295 Decimal expansion of 6^(6^(6^6)) = 6^^4.

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A241296 Decimal expansion of 7^(7^(7^7)) = 7^^4.

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