

A008952


Leading digit of 2^n.


12



1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 3, 6, 1, 2, 4, 9, 1, 3, 7, 1, 3, 6, 1, 2, 4, 9, 1, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Statistically, sequence obeys Benford's law, i.e. digit d occurs with probability log_10(1 + 1/d); thus 1 appears about 6.6 times more often than 9.  Lekraj Beedassy, May 04 2005
The most significant digits of the nth powers of 2 are not cyclic and in the first 1000000 terms, 1 appears 301030 times, 2 appears 176093, 3 appears 124937, 4 appears 96911, 5 appears 79182, 6 appears 66947, 7 appears 57990, 8 appears 51154 and 9 appears 45756 times.  Robert G. Wilson v, Feb 03 2008
In fact the sequence follows Benford's law precisely by the equidistribution theorem.  Charles R Greathouse IV, Oct 11 2015


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..100000.
J. Derbyshire, Lead Digit of 2 to the Power of N
Brady Haran and Dmitry Kleinbock, Powers of 2, Numberphile video (2015). More footage
R. Shiquan, The first digit of 2^n, vixra:1210.0007
Wikipedia, Frank Benford's law.
Wikipedia, George Zipf's law.
Index entries for sequences related to Benford's law


FORMULA

a(n) = [2^n / 10^([log_10(2^n)])] = [2^n / 10^([n*log_10(2)])].
a(n) = A000030(A000079(n)).  Omar E. Pol, Jul 04 2019


MATHEMATICA

a[n_] := First@ IntegerDigits[2^n]; Array[a, 105, 0] (* Robert G. Wilson v, Feb 03 2008 and corrected Nov 24 2014 *)


PROG

(PARI) a(n)=digits(2^n)[1] \\ Charles R Greathouse IV, Oct 11 2015


CROSSREFS

Cf. A000030, A000079.
Sequence in context: A023104 A133145 A317414 * A268516 A021407 A131609
Adjacent sequences: A008949 A008950 A008951 * A008953 A008954 A008955


KEYWORD

nonn,base


AUTHOR

Leonid Broukhis


STATUS

approved



