A261009 Write 2^n in base 3, add up the "digits".

1, 2, 2, 4, 4, 4, 4, 6, 4, 8, 8, 10, 10, 8, 10, 16, 12, 14, 12, 16, 14, 18, 16, 12, 10, 12, 14, 20, 20, 22, 24, 26, 24, 22, 22, 22, 18, 20, 26, 28, 28, 28, 26, 30, 30, 30, 26, 26, 26, 32, 38, 40, 38, 38, 28, 34, 40, 42, 38, 40, 46, 40, 38, 42, 48, 44, 42, 40, 42, 48, 48, 44

Offset

0,2

Comments

Comment from Jean-Paul Allouche, Oct 25 2015: As mentioned by Holdum et al. (2015) the following problem, cited in "Concrete Mathematics" by Graham, Knuth, and Patashnik (1994), is still open: prove that for all n > 256, binomial(2n,n) is either divisible by 4 or by 9 (cf. A000984). This can be easily reduced to show that, for all k >= 9, 2*a(k) - a(k+1) >= 4. This has been proved up to huge values of k (Holdum et al. mention k = 10^{13}).

For additional information about the divisibility of binomial(2n,n) by squares see the comments and references in A000984, - N. J. A. Sloane, Oct 29 2015

Links

Giovanni Resta, Table of n, a(n) for n = 0..10000

Cernenoks J., Iraids J., Opmanis M., Opmanis R., Podnieks K., Integer complexity: experimental and analytical results II, arXiv:1409.0446 [math.NT] (September 2014)

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994

Sebastian Tim Holdum, Frederik Ravn Klausen, Peter Michael Reichstein Rasmussen, Powers in prime bases and a problem on central coefficients, Integers 15 (2015), #A43

K. Podnieks, Digits of pi: limits to the seeming randomness, arXiv:1411.3911 [math.NT], 2014.

Formula

a(n) = A053735(A000079(n)). - Michel Marcus, Aug 14 2015

Example

2^7 = 128_10 = 11202_3, so a(7) = 1+1+2+0+2 = 6.

Maple

S:=n->add(i, i in convert(2^n, base, 3)); [seq(S(n), n=0..100)];

Mathematica

Table[Total@ IntegerDigits[2^n, 3], {n, 0, 100}] (* Giovanni Resta, Aug 14 2015 *)

Prog

(PARI) a(n) = vecsum(digits(2^n, 3)); \\ Michel Marcus, Aug 14 2015
(Haskell)
a261009 = a053735 . a000079  -- Reinhard Zumkeller, Aug 14 2015

Crossrefs

Cf. A000079, A000984, A053735, A007089.

Sum of digits of k^n in base b for various pairs (k,b): A001370 (2,10), A011754 (3,2), A261009 (2,3), A261010 (5,3).

Sequence in context: A080374 A165040 A035683 * A239896 A001670 A100144

Adjacent sequences: A261006 A261007 A261008 * A261010 A261011 A261012

Keyword

nonn,base

Author

N. J. A. Sloane, Aug 14 2015

Status

approved