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A209229
Characteristic function of powers of 2, cf. A000079.
175
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.
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).
Sequence in context: A369961 A219189 A029691 * A365089 A295890 A342704
KEYWORD
nonn,mult,easy
AUTHOR
Reinhard Zumkeller, Mar 06 2012
STATUS
approved