A209229 Characteristic function of powers of 2, cf. A000079.

0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Reinhard Zumkeller, Mar 06 2012

Keywords

• nonn
• mult
• easy

Comments

Essentially the same as A036987 (the Fredholm-Rueppel sequence).

Completely multiplicative with a(2^e) = 1, a(p^e) = 0 for odd primes p. - Mitch Harris, Apr 19 2005

Moebius transform of A001511. - R. J. Mathar, Jun 20 2014

Examples

			x + x^2 + x^4 + x^8 + x^16 + x^32 + x^64 + x^128 + x^256 + x^512 + x^1024 + ...
		

References

• Michel Dekking, Michel Mendes France and Alf van der Poorten, "Folds", The Mathematical Intelligencer, Vol. 4, No. 3 (1982), pp. 130-138 & front cover, and Vol. 4, No. 4 (1982), pp. 173-181 (printed in two parts).
• Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537 (terms 0..1000 from G. C. Greubel)
• Index entries for characteristic functions
• Index to divisibility sequences

Crossrefs

Cf. A001511, A029837 (partial sums), A087003 (moebius transform), A090678, A104977, A154955 (Dirichlet inverse).

Cf. A069517, A154269, A201219, A255738.

Programs

C

int a (unsigned long n) { return n & !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */

Haskell

a209229 n | n < 2 = n
          | n > 1 = if m > 0 then 0 else a209229 n'
          where (n',m) = divMod n 2

Maple

A209229 := proc(n)
    if n <= 0 then
        0 ;
    elif n = 1 then
        1;
    elif type (n,'odd') or A001221(n) > 1 then
        0 ;
    else
        1;
    end if;
end proc:
seq(A209229(n),n=0..40) ; # R. J. Mathar, Jan 07 2021

Mathematica

a[n_] := Boole[n == 2^IntegerExponent[n, 2]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 06 2014 *)
Table[If[IntegerQ[Log[2,n]],1,0],{n,0,100}] (* Harvey P. Dale, Jun 24 2018 *)

PARI

a(n)=n==1<Charles R Greathouse IV, Mar 07 2012

PARI

{a(n) = if( n<2 || n%2, n==1, isprimepower(n) > 0)} \\ Michael Somos, Jan 03 2013

Python

def A209229(n): return int(not(n&-n)^n) if n else 0 # Chai Wah Wu, Jul 08 2022

Formula

a(A000079(n)) = 1; a(A057716(n)) = 0.

a(n+1) = A036987(n).

a(n) = if n < 2 then n else (if n is even then a(n/2) else 0).

The generating function g(x) satisfies g(x) - g(x^2) = x. - Joerg Arndt, May 11 2010

Dirichlet g.f.: 1/(1 - 2^(-s)). - R. J. Mathar, Mar 07 2012

G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x / (1 + x / (1 - x / ...)))))) = x / (1 + b(1) * x / (1 + b(2) * x / (1 + b(3) * x / ...))) where b(n) = (-1)^ A090678(n+1). - Michael Somos, Jan 03 2013

With a(0) = 0 removed is convolution inverse of A104977. - Michael Somos, Jan 03 2013

From Antti Karttunen, Nov 19 2017: (Start)

a(n) = abs(A154269(n)).

For n > 1, a(n) = A069517(n)/2 = 2 - A201219(n). (End)

a(n) = A048298(n)/n. - R. J. Mathar, Jan 07 2021

a(n) = floor((2^n)/n) - floor((2^n - 1)/n), for n>=1. - Ridouane Oudra, Oct 15 2021