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

0,4

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.

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.

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

Example

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.

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]

Crossrefs

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

Sequence in context: A360439 A115064 A086868 * A090269 A086867 A090266

Adjacent sequences: A241296 A241297 A241298 * A241300 A241301 A241302

Keyword

nonn,base,easy,changed

Author

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

Status

approved