Search: id:A010052
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%I A010052
%S A010052 1,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T A010052 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A010052 0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0
%N A010052 Characteristic function of squares: 1 if n is a square else 0.
%C A010052 For n >= 1 another formula for a(n) is: a(n) = d(n) mod 2 where d(n) 
               is the number of divisors of n, A000005. - Ahmed Fares (ahmedfares(AT)my-deja.com), 
               Apr 19 2001
%C A010052 G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=(u-w)^2-(v-w)(v+w-1) 
               - Michael Somos, Jul 19 2004
%C A010052 Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Mar 15 2009: 
               (Start)
%C A010052 =Decimal expansion of Sum(n=1...inf), 1/(2^n)^((i^4)*n) * 1/(5^n)^((i^4)*n)
%C A010052 = (1/(2^n)*1/(5^n))^(i^4*n) = (1/(10^n))^(i^4*n)
%C A010052 (End)
%C A010052 =(1/(10^n))^n [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Mar 15 2009]
%D A010052 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 
               2003, p. 4.
%D A010052 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 48, Problem 20.
%H A010052 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%H A010052 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A010052 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A010052 a(n) = [sqrt(n)] - [sqrt(n-1)] (n>0).
%F A010052 Dirichlet generating function: zeta(2s). - Franklin T. Adams-Watters, 
               Sep 11 2005.
%F A010052 G.f. (theta_3(0,x) + 1)/2, where theta_3 is a Jacobi theta function. 
               - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 19 2006
%F A010052 a(n) = f(n,0) with f(x,y) = if x>0 then f(x-2*y-1,y+1) else 0^(-x). [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 26 2008]
%F A010052 For n>=1 a(n)=sumdiv(n,d,(-1)^bigomega(d)) [From Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 25 2009]
%F A010052 a(n) <= A093709(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 14 2009]
%p A010052 readlib(issqr): f := i->if issqr(i) then 1 else 0; fi; [ seq(f(i),i=0..100) 
               ];
%o A010052 (PARI) a(n)=issquare(n)
%o A010052 (PARI) a(n)=if(n<1,1,sumdiv(n,d,(-1)^bigomega(d))) [From Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Oct 25 2009]
%Y A010052 Cf. A008836.
%Y A010052 Sequence in context: A127692 A014305 A023533 this_sequence A039985 A127239 
               A129186
%Y A010052 Adjacent sequences: A010049 A010050 A010051 this_sequence A010053 A010054 
               A010055
%K A010052 nonn,nice,easy,mult,new
%O A010052 0,1
%A A010052 N. J. A. Sloane (njas(AT)research.att.com).
%E A010052 More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 19 
               2006

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