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

0,1

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

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

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

Example

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

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 *)

Prog

(Haskell)
a209229 n | n < 2 = n
          | n > 1 = if m > 0 then 0 else a209229 n'
          where (n', m) = divMod n 2
(PARI) a(n)=n==1<<valuation(n, 2) \\ 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
(C) int a (unsigned long n) { return n & !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */
(Python)
def A209229(n): return int(not(n&-n)^n) if n else 0 # Chai Wah Wu, Jul 08 2022

Crossrefs

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

Cf. A069517, A154269, A201219, A255738.

Sequence in context: A189135 A219189 A029691 * A365089 A295890 A342704

Adjacent sequences: A209226 A209227 A209228 * A209230 A209231 A209232

Keyword

nonn,mult,easy

Author

Reinhard Zumkeller, Mar 06 2012

Status

approved