A001370 Sum of digits of 2^n.

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, 29, 40, 35, 43, 41, 37, 47, 58, 62, 61, 59, 64, 56, 67, 71, 61, 50, 46, 56, 58, 62, 70, 68, 73, 65, 76, 80, 79, 77, 82, 92, 85, 80, 70, 77

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• base
• easy
• nonn

Comments

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

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

The Radcliffe preprint shows that a(n) > log_4(n). - M. F. Hasler, May 18 2017

Sierpiński shows that if n >= A137284(k-1) then a(n) >= k (Problem 209). - David Radcliffe, Dec 26 2022

References

• Archimedeans Problems Drive, Eureka, 26 (1963), 12.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Zak Seidov, Table of n, a(n) for n = 0..10000
• David Radcliffe, The growth of digital sums of powers of two. Preprint, 2015.
• David G. Radcliffe, The growth of digital sums of powers of two, arXiv:1605.02839 [math.NT], 2016.
• W. Sierpiński, 250 Problems in Elementary Number Theory, 1970.
• C. L. Stewart, On the representation of an integer in two different bases, Journal für die reine und angewandte Mathematik 319 (1980): 63-72.

Crossrefs

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).

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

Programs

Haskell

a001370 = a007953 . a000079  -- Reinhard Zumkeller, Aug 14 2015

Maple

seq(convert(convert(2^n,base,10),`+`),n=0..1000); # Robert Israel, Mar 29 2015

Mathematica

Table[Total[IntegerDigits[2^n]], {n, 0, 55}] (* Vincenzo Librandi, Oct 08 2013 *)

PARI

a(n)=sumdigits(2^n); \\ Michel Marcus, Nov 01 2013

Python

[sum(map(int, str(2**n))) for n in range(56)] # David Radcliffe, Mar 29 2015

Formula

a(n) = A007953(A000079(n)). - Michel Marcus, Nov 01 2013