A001370 Sum of digits of 2^n. (Formerly M1085 N0414)

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

0,2

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.

Formula

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

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 *)

Prog

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

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.

Sequence in context: A225746 A021406 A065075 * A195715 A343629 A039794

Adjacent sequences: A001367 A001368 A001369 * A001371 A001372 A001373

Keyword

base,easy,nonn

Author

N. J. A. Sloane, Simon Plouffe

Status

approved