A241291 -id:A241291 - cp's OEIS frontend

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

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))).

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula