%I M1085 N0414 #91 Dec 27 2022 08:33:34

%S 1,2,4,8,7,5,10,11,13,8,7,14,19,20,22,26,25,14,19,29,31,26,25,41,37,

%T 29,40,35,43,41,37,47,58,62,61,59,64,56,67,71,61,50,46,56,58,62,70,68,

%U 73,65,76,80,79,77,82,92,85,80,70,77

%N Sum of digits of 2^n.

%C Same digital roots as A065075 (sum of digits of the sum of the preceding numbers) and A004207 (sum of digits of all previous terms); they enter into the cycle {1 2 4 8 7 5}. - _Alexandre Wajnberg_, Dec 11 2005

%C It is believed that a(n) ~ n*9*log_10(2)/2, but this is an open problem. - _N. J. A. Sloane_, Apr 21 2013

%C The Radcliffe preprint shows that a(n) > log_4(n). - _M. F. Hasler_, May 18 2017

%C Sierpiński shows that if n >= A137284(k-1) then a(n) >= k (Problem 209). - _David Radcliffe_, Dec 26 2022

%D Archimedeans Problems Drive, Eureka, 26 (1963), 12.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Zak Seidov, <a href="/A001370/b001370.txt">Table of n, a(n) for n = 0..10000</a>

%H David Radcliffe, <a href="/A001370/a001370_1.pdf">The growth of digital sums of powers of two</a>. Preprint, 2015.

%H David G. Radcliffe, <a href="http://arxiv.org/abs/1605.02839">The growth of digital sums of powers of two</a>, arXiv:1605.02839 [math.NT], 2016.

%H W. Sierpiński, <a href="https://www.isinj.com/mt-usamo/250%20Problems%20in%20Elementary%20Number%20Theory%20-%20Sierpinski%20(1970).pdf">250 Problems in Elementary Number Theory</a>, 1970.

%H C. L. Stewart, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002197707">On the representation of an integer in two different bases</a>, Journal für die reine und angewandte Mathematik 319 (1980): 63-72.

%F a(n) = A007953(A000079(n)). - _Michel Marcus_, Nov 01 2013

%p seq(convert(convert(2^n,base,10),`+`),n=0..1000); # _Robert Israel_, Mar 29 2015

%t Table[Total[IntegerDigits[2^n]], {n, 0, 55}] (* _Vincenzo Librandi_, Oct 08 2013 *)

%o (PARI) a(n)=sumdigits(2^n); \\ _Michel Marcus_, Nov 01 2013

%o (Python) [sum(map(int, str(2**n))) for n in range(56)] # _David Radcliffe_, Mar 29 2015

%o (Haskell)

%o a001370 = a007953 . a000079 -- _Reinhard Zumkeller_, Aug 14 2015

%Y Cf. sum of digits of k^n: A004166 (k=3), A065713 (k=4), A066001(k=5), A066002 (k=6), A066003(k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12).

%Y Cf. A007953, A000079, A261009, A011754, A137284.

%K base,easy,nonn

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_