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

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

Author

Marco Ripà, Mar 03 2023

Keywords

• nonn
• cons
• easy
• fini
• full

Comments

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...

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Python

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

Formula

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.