Robert Munafo - cp's OEIS frontend

A278835 Prime factors (counting multiplicity) of 10^10^10^10^2 - 1.

3, 3, 11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, 1201, 1409, 1601, 3541, 4001, 4801, 5051, 9091, 10753, 15361, 16001, 19841, 21001, 21401, 24001, 25601, 27961, 37501, 40961, 43201, 60101, 62501, 65537, 69857, 76001, 76801, 160001, 162251, 163841, 307201, 453377, 524801, 544001, 670001, 952001, 976193, 980801

Offset: 1

Robert Munafo and Robert G. Wilson v, Nov 28 2016

From Jon E. Schoenfield, Dec 02 2016, paraphrasing information from the Munafo link: (Start)
The decimal expansion of 10^10^10^10^2 - 1 would be 1 googolplex digits long, with each digit a 9. Many factors of this number can be identified using simple facts of modular arithmetic.
Since its digits are all 9's, it is divisible by 9=3*3. Since its digits are all 9's and the number of digits is even, it is divisible by 99 (as are 9999=99*101, 999999=99*10101, 99999999=99*1010101, etc.), and thus divisible by 11.
By the same principle, it is divisible by 9999, 99999, 99999999, and by any other number whose decimal expansion consists of k 9's where k is of the form 2^a * 5^b, where a and b are nonnegative integers up to 10^100 (see A003592) and all their divisors. Additional factors can be found using Fermat's Little Theorem.
Consequently, a large number of factors of 10^10^10^10^2 - 1 are known. (End)

			10^10^10^10^2 - 1 = 10^10^10^100 - 1 = 999...999 (a total of a googolplex of nines).

Robert G. Wilson v, Table of n, a(n) for n = 1..587
Dario Alejandro Alpern, Known prime factors of Googolduplex - 1
Dario Alejandro Alpern, Known 1-digit prime factors of Googolduplex - 1
Robert P. Munafo, Notable Properties of Specific Numbers

Cf. A227246.

A272853 Ramanujan's alpha-series.

Original entry on oeis.org

9, 791, 65601, 5444135, 451797561, 37493753471, 3111529740489, 258219474707159, 21429104870953665, 1778357484814447079, 147582242134728153849, 12247547739697622322431

Offset: 0

Robert Munafo, May 08 2016

nonn

Ramanujan's notes define this by the same G.f. as A051028 (the a-series) but using Laurent series expansion. These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n). They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).

			a(3)=5444135 because 5444135^3 + 5593538^3 = 6954572^3 - 1.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).

Seiichi Manyama, Table of n, a(n) for n = 0..520
Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

Cf. A051028, A051029, A051030.

Cf. A272854, A272855.

Rest@ CoefficientList[Normal@ Series[(1 + 53*a + 9*a^2)/(1 - 82*a - 82*a^2 + a^3), {a, Infinity, 20}], 1/a] (* Giovanni Resta, May 08 2016 *)

G.f.: (9+53*x+x^2)/(1-82*x-82*x^2+x^3).
a(-3)=-11161; a(-2)=-135; a(-1)=-1; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.

A272854 Ramanujan's beta-series.

Original entry on oeis.org

10, 812, 67402, 5593538, 464196268, 38522696690, 3196919629018, 265305806511788, 22017185020849402, 1827161050923988562, 151632350041670201260, 12583657892407702716002

Offset: 0

Robert Munafo, May 08 2016

Ramanujan's notes define this by the same G.f. as A051030 (the c-series) but using Laurent series expansion. It is mislabeled as "gamma" in Ramanujan's notes. These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n). They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).

			a(3)=5593538 because 5444135^3 + 5593538^3 = 6954572^3 - 1.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).

Seiichi Manyama, Table of n, a(n) for n = 0..520
Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

Cf. A051028, A051029, A051030.

Cf. A272853, A272855.

Rest@ CoefficientList[ Normal@Series[-(2 + 8*x - 10*x^2)/(1 - 82*x - 82*x^2 + x^3), {x, Infinity, 20}], 1/x] (* Giovanni Resta, May 08 2016 *)

G.f.: (10-8*x-2*x^2)/(1-82*x-82*x^2+x^3).
a(-3)=14258; a(-2)=172; a(-1)=2; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.

A272855 Ramanujan's gamma-series.

Original entry on oeis.org

12, 1010, 83802, 6954572, 577145658, 47896135058, 3974802064140, 329860675188578, 27374461238587818, 2271750422127600332, 188527910575352239722, 15645544827332108296610

Offset: 0

Robert Munafo, May 08 2016

nonn

Ramanujan's notes define this by the same G.f. as A051029 (the b-series) but using Laurent series expansion. It is mislabeled as "beta" in Ramanujan's notes. These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n). They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).

			a(3)=6954572 because 5444135^3 + 5593538^3 = 6954572^3 - 1.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).

Seiichi Manyama, Table of n, a(n) for n = 0..520
Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

Cf. A051028, A051029, A051030.

Cf. A272853, A272854.

Rest@ CoefficientList[ Normal@ Series[-1*(2 - 26 a - 12 a^2)/(1 - 82*a - 82*a^2 + a^3), {a, Infinity, 10}], 1/a] (* Giovanni Resta, May 08 2016 *)

G.f.: x*(12+26*x-2*x^2)/(1-82*x-82*x^2+x^3).
a(-3)=11468; a(-2)=138; a(-1)=2; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.

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

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

Original entry on oeis.org

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

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

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

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

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

cp's OEIS Frontend

User: Robert Munafo

A278835 Prime factors (counting multiplicity) of 10^10^10^10^2 - 1.

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A272853 Ramanujan's alpha-series.

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A272854 Ramanujan's beta-series.

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A272855 Ramanujan's gamma-series.

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

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PARI

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

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