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

Author

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

Keywords

• nonn
• base
• easy

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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]

Formula

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