|
%I #54 Jul 07 2022 20:06:09
%S 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,
%T 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,
%U 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
%N Leading digit of 2^n.
%C 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
%C The most significant digits of the n-th 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
%C In fact the sequence follows Benford's law precisely by the equidistribution theorem. - _Charles R Greathouse IV_, Oct 11 2015
%H Robert G. Wilson v, <a href="/A008952/b008952.txt">Table of n, a(n) for n = 0..100000</a>.
%H Brady Haran and Dmitry Kleinbock, <a href="https://www.youtube.com/watch?v=pKwsPBeSiOc">Powers of 2</a>, Numberphile video (2015). <a href="https://www.youtube.com/watch?v=vetsor9NTF8">More footage</a>.
%H Jaap Spies, <a href="http://www.jaapspies.nl/bookb5.pdf">A Bit of Math, The Art of Problem Solving</a>, Jaap Spies Publishers (2019).
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Benford%27s_law">Benford's law</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Zipf's_law">Zipf's law</a>.
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F a(n) = [2^n / 10^([log_10(2^n)])] = [2^n / 10^([n*log_10(2)])].
%F a(n) = A000030(A000079(n)). - _Omar E. Pol_, Jul 04 2019
%p a:= n-> parse(""||(2^n)[1]):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Aug 06 2021
%t a[n_] := First@ IntegerDigits[2^n]; Array[a, 105, 0] (* _Robert G. Wilson v_, Feb 03 2008 and corrected Nov 24 2014 *)
%o (PARI) a(n)=digits(2^n)[1] \\ _Charles R Greathouse IV_, Oct 11 2015
%o (Python)
%o def A008952(n): return int(str(1<<n)[0]) # _Chai Wah Wu_, Jul 07 2022
%Y Cf. A000030, A000079.
%K nonn,base
%O 0,2
%A _Leonid Broukhis_
|
