A241299 -id:A241299 - cp's OEIS frontend

A364789 Initial digit of (n^n)^n (A002489(n)).

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

Offset: 0

Marco Ripà, Aug 08 2023

a(0) = 1 is from (0^0)^0 = 1 per A002489.

The author conjectures that this sequence obeys the well-known Benford's law.

			a(4) = 4, since (4^4)^4 = 4^(4^2) = 4294967296.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Pointless Large numbers stuff by Cookiefonster, 2.03 The Weak Hyper-Operators.
Wikipedia, Benford's law.

Cf. A000030, A002489, A241299, A363746, A364837, A364855.

Cf. A229522 (final digit).

A364789[n_] := If[n == 0, 1, First[IntegerDigits[(n^n)^n]]];
Array[A364789, 100, 0] (* Paolo Xausa, Jan 31 2024 *)

def A364789(n): return int(str((n**n)**n)[0]) # Chai Wah Wu, Aug 10 2023

a(n) = floor(((n^n)^n)/10^floor(log_10((n^n)^n))).

a(n) = A000030(A002489(n)).

A362004 Initial digit of the decimal expansion of the tetration 2^^n (in Don Knuth's up-arrow notation).

Original entry on oeis.org

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

Offset: 0

Marco Ripà, Apr 02 2023

The most significant digit of the base-10 representation of 2^(2^(2^...)) n times is given by floor(2^^n/10^(len(2^^n)-1)), where len(2^^n) indicates the number of digits of the argument.

Although it is known that 2^^6 starts with the digit 2 (see A241291), a(7) is not currently known (for details, see Googology link, section "First digits in tetration").

			a(4) = 6, since 2^^4 = 65536.

Googology, Tetration.
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Cf. A241291, A241299, A244059.

a(n) = floor(2^^n/10^floor(log(2^^n))).

A246564 The n-th least-significant decimal digit of n^^n (in Don Knuth's up-arrow notation).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 30 2014

This sequence was inspired by the 41st Wohascum County problem.
The distribution of the first 500 terms beginning with 0: 101, 43, 40, 42, 29, 49, 43, 53, 58, 42.
The distribution does not conform to Benford's / Zipf's law, but seems to be evenly distributed once multiples of ten are excluded.

George T. Gilbert, Mark I. Krusemeyer and Loren C. Larson, The Wohascum County Problem Book, The Mathematical Association of America, Dolciani Mathematical Expositions No. 14, 1993, problem 41 "What is the fifth digit from the end (the ten thousands digit) of the number 5^5^5^5^5?", page 11 and solution on page 76.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Robert P. Munafo, Sequence A094358, 2^^N = 1 mod N.
Robert P. Munafo, Hyper4 Iterated Exponential Function.
Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi.
Wikipedia, Knuth's up-arrow notation.
Index entries for sequences related to Benford's law

Cf. A241293, A241299, A244059.

(* first load "SuperPowerMod" from Vardi, see link above, and then *) f[n_] := Quotient[ SuperPowerMod[ n, n, 10^n], 10^(n - 1)]; Array[f, 105]

if n (mod 10) == 0 then a(n) = 0.

A361100 Decimal expansion of 2^(2^(2^(2^2))) = 2^^5.

Original entry on oeis.org

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

Offset: 19730

Marco Ripà, Mar 03 2023

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
so that 2^2^2^2^2 = 2^^5 = 2^(2^(2^(2^2))) = 2^65536 = 20035299304068464649790723515602557504478254755697514192650169737108940595563...

			2003529930406846464979072351560255750447825475569751419265016973710894059556311453089506130880933348
(...19529 digits omitted...)
5775699146577530041384717124577965048175856395072895337539755822087777506072339445587895905719156736.
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.

Chai Wah Wu, Table of n, a(n) for n = 19730..39458
Googology Wiki, Tetration.
Pointless Large numbers stuff by Cookiefonster, 2.02 Knuth's Up-Arrows and the Hyper-Operators.
Robert P. Munafo, Sequence A094358, 2^^N = 1 mod N.

Cf. A085667, A014221, A241291, A241292, A241293, A241294, A241295, A241296, A241297, A241298, A241299, A243913.

nbrdgt = 100; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[2, 2^2^2^2]
IntegerDigits[2^65536][[;;100]] (* Paolo Xausa, Jan 31 2024 *)

def A361100(n): return (1<<(1<<(1<<(1<<(1<<1)))))//10**(39458-n)%10 # Chai Wah Wu, Apr 03 2023

Equals 2^2^2^2^2 = 2^^5 = (((((((((((((((2^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2.

A363746 Initial digit of the decimal expansion of the tetration n^^n (in Don Knuth's up-arrow notation).

Original entry on oeis.org

1, 1, 4, 7, 2

Offset: 0

Marco Ripà, Jun 19 2023

a(5), the most significant digit of the tetration 5^^5, has been estimated to be equal to 1 (and this is also consistent with Benford's law), but there is not any strict proof at the present time and computers are not powerful enough to calculate it without uncertainty.

			a(3) = 7 since 3^^3 = 7625597484987.

Allam's Numbers, Digits of tetrational numbers
A. Bogomolny, Benford's Law and Zipf's Law, Cut the Knot.org.
Googology, Tetration.
Eric Weisstein's World of Mathematics, Joyce Sequence.
Wikipedia, Knuth's up-arrow notation.

Cf. A000030, A004231 (n^^n), A241293 (4^^4 digits).

Cf. A241299, A244059, A362004.

a(n) = floor(n^^n/10^floor(log_10(n^^n))).

a(n) = A000030(A004231(n)).

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A364789 Initial digit of (n^n)^n (A002489(n)).

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A362004 Initial digit of the decimal expansion of the tetration 2^^n (in Don Knuth's up-arrow notation).

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A246564 The n-th least-significant decimal digit of n^^n (in Don Knuth's up-arrow notation).

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A361100 Decimal expansion of 2^(2^(2^(2^2))) = 2^^5.

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A363746 Initial digit of the decimal expansion of the tetration n^^n (in Don Knuth's up-arrow notation).

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