A241294 -id:A241294 - cp's OEIS frontend

A241299 Initial digit of the decimal expansion of n^(n^n) or n^^3 (in Don Knuth's up-arrow notation).

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

Offset: 0

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

0^^3 = 0 since 0^^k = 1 for even k, 0 for odd k, k >= 0.
Conjecture: the distribution of the initial digits obey Zipf's law.
The distribution of the first 1000 terms beginning with 1: 302, 196, 124, 91, 72, 46, 71, 53, 45.

			a(0) = 0, a(1) = 1, a(2) = 1 because 2^(2^2) = 16, a(3) = 7 because 3^(3^3) = 7625597484987 and its initial digit is 7, etc.

Robert P. Munafo and Robert G. Wilson v, Table of n, a(n) for n = 0..1000
Cut the Knot.org, Benford's Law and Zipf's Law, A. Bogomolny, Zipf's Law, Benford's Law from Interactive Mathematics Miscellany and Puzzles.
Hans Havermann, Next 5 terms.
M. E. J. Newman, Power laws, Pareto distributions and Zipf's law.
Eric Weisstein's World of Mathematics, Joyce Sequence.
Wikipedia, Knuth's up-arrow notation.
Wikipedia, Zipf's law.
Index entries for sequences related to Benford's law.

Cf. A000030, A000312, A002488, A066022, A241291, A241292, A241293, A241294, A241295, A241296, A241297, A241298.

g[n_] := Quotient[n^p, 10^(Floor[ p*Log10@ n] - (1004 + p))]; f[n_] := Block[{p = n}, Quotient[ Nest[ g@ # &, p, p], 10^(1004 + p)]]; Array[f, 105, 0]

For n > 0, a(n) = floor(t/10^floor(log_10(t))) where t = n^(n^n).

a(n) = A000030(A002488(n)). - Omar E. Pol, Jul 04 2019

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

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

A243913 Decimal expansion of 9^(9^(9^9)) = 9^^4.

Original entry on oeis.org

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

Offset: 1

Hans Havermann and Robert G. Wilson v, Jun 14 2014

The offset is taken to be 1 because the true offset would be 4.085349171835445 * 10^369693099, too large to be written out in full.

			=2141983294796805611333364373442480830147227072845128488706516195982808749656704847036118447249917368...(4.085349171835445... * 10^369693099) ... 3771540670946945552331518959254852001991324340257630363975097419408973491530163140828233401045865289.
The first and last 100 digits are shown above, with the intervening digits omitted. The final one hundred digits cannot be computed with: PowerMod[9, 9^9^9, 10^100] with any version of Mathematica before version 9. Instead (* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file and then *) SuperPowerMod[9, 4, 10^100].

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

Hans Havermann and Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Robert P. Munafo, Hyper4 Iterated Exponential Function
Robert P. Munafo and Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi
Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi

Cf. A002488, A054382, A085667, A202955, A054382, A014221, A241291, A241292, 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[ 9, 9^9^9] (* needs version 9.0 to run *)

9^(9^(9^9)) = ((((( ... 387420478 ... (((((9^9)^9)^9)^9)^9) ... 387420478 ... ^9)^9)^9)^9)^9)^9.

Keyword: fini added by Jianing Song, Sep 18 2019

A244059 Initial digit of the decimal expansion of n^(n^(n^n)) or n^^4 (in Don Knuth's up-arrow notation).

Original entry on oeis.org

1, 1, 6, 1, 2, 1, 4, 7, 6, 2, 1

Offset: 0

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

This sequence can also be written as (n↑↑4) in Knuth up-arrow notation.
0^^4 = 1 since 0^^k = 1 for even k, 0 for odd k, k >= 0.
Conjecture: the distribution of the initial digits obey G. K. Zipf's law.

			a(4)=2 because A241293(1)=2.

Cut the Knot.org, Benford's Law and Zipf's Law, A. Bogomolny, Zipf's Law, Benford's Law from Interactive Mathematics Miscellany and Puzzles.
M. E. J. Newman, Power laws, Pareto distributions and Zipf's law.
Eric Weisstein's World of Mathematics, Joyce Sequence
Wikipedia, Knuth's up-arrow notation
Wikipedia, Zipf's law
Index entries for sequences related to Benford's law

Cf. A241291, A241292, A241293, A241294, A241295, A241296, A241297, A243913, A241299.

Cf. A324220 (number of digits).

a(n)=digits(n^n^n^n)[1] \\ impractical for large n; Charles R Greathouse IV, May 13 2015

cp's OEIS Frontend

A241299 Initial digit of the decimal expansion of n^(n^n) or n^^3 (in Don Knuth's up-arrow notation).

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

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