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A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

%I A065333 #54 Dec 20 2024 12:32:39
%S A065333 1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,
%T A065333 0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%U A065333 0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0
%N A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).
%C A065333 Dirichlet inverse of b(n) where b(n) = 0 except for: b(1) = b(6) = -b(2) = -b(3) = 1. - _Alexander Adam_, Dec 26 2012
%H A065333 Reinhard Zumkeller, <a href="/A065333/b065333.txt">Table of n, a(n) for n = 1..10000</a>
%H A065333 A. Pakapongpun and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ward/ward17.html">Functorial Orbit counting</a>, JIS 12 (2009) 09.2.4, example 9.
%H A065333 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A065333 a(n) = if n = A003586(k) for some k then 1 else 0.
%F A065333 a(n) = signum(A065332(n)), where signum = A057427.
%F A065333 a(n) = if A065330(n) = 1 then 1 else 0 = 1 - signum(A065330(n) - 1).
%F A065333 a(n) = Product_{p prime and p|n} 0^floor(p/4). - _Reinhard Zumkeller_, Nov 19 2004
%F A065333 Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = 0 for prime p > 3. Dirichlet g.f. 1/(1-2^-s)/(1-3^-s). - _Franklin T. Adams-Watters_, Sep 01 2006
%F A065333 a(n) = 0^(A038502(A000265(n)) - 1). - _Reinhard Zumkeller_, Sep 28 2008
%F A065333 a(n) = Sum_{d|n} mu(6*d). - _Benoit Cloitre_, Oct 18 2009
%t A065333 a[n_] := Boole[ 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] == n]; Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, May 16 2013, after _Charles R Greathouse IV_ *)
%o A065333 (PARI) a(n)=sumdiv(n,d,moebius(6*d)) \\ _Benoit Cloitre_, Oct 18 2009
%o A065333 (PARI) a(n)=3^valuation(n,3)<<valuation(n,2)==n \\ _Charles R Greathouse IV_, Aug 21 2011
%o A065333 (Haskell)
%o A065333 a065333 = fromEnum . (== 1) . a038502 . a000265
%o A065333 -- _Reinhard Zumkeller_, Jan 08 2013, Apr 12 2012
%o A065333 (Python)
%o A065333 from sympy import multiplicity
%o A065333 def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m) # _Chai Wah Wu_, Dec 20 2024
%Y A065333 Characteristic function of A003586.
%Y A065333 Cf. A000265, A007814, A007949, A038502, A065330, A065332, A071521 (partial sums), A072078 (inverse Möbius transform).
%K A065333 mult,nonn,easy
%O A065333 1,1
%A A065333 _Reinhard Zumkeller_, Oct 29 2001