%I A000196
%S A000196 0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,
%T A000196 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,
%U A000196 8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10
%N A000196 Integer part of square root of n. Or, number of squares <= n. Or, n appears 
               2n+1 times.
%C A000196 Also the integer part of the geometric mean of the divisors of n. - Amarnath 
               Murthy (amarnath_murthy(AT)yahoo.com), Dec 19 2001
%C A000196 a(n)=Card(k, 0<k<=n such that k is relatively prime to core(k)) where 
               core(x) is the square-free part of x. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 02 2002
%C A000196 Number of numbers k (<=n) with an odd number of divisors - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Sep 07 2002
%D A000196 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 73, problem 23.
%D A000196 K. Atanassov, On the 100-th, 101-st and the 102-th Smarandache Problems, 
               Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, 
               Vol. 5 (1999), No. 3, 94-96.
%D A000196 K. Atanassov, On Some of Smarandache's Problems, American Research Press, 
               1999, 16-21.
%D A000196 N. J. A. Sloane and A. R. Wilks, On sequences of Recaman type, paper 
               in preparation, 2006.
%D A000196 F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 
               1993.
%H A000196 Franklin T. Adams-Watters, <a href="b000196.txt">Table of n, a(n) for 
               n = 0..10000</a>
%H A000196 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">
               On Some of Smarandache's Problems</a>
%H A000196 H. Bottomley, <a href="a000196.gif">Illustration of A000196, A048760, 
               A053186</a>
%H A000196 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">
               Only Problems, Not Solutions!</a>.
%F A000196 a(n) = a(n-1) + floor(n/(a(n-1)+1)^2), a(0) = 0. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Apr 12 2004
%F A000196 a(n)=sum{0<k<=n, A010052(k)}. G.f.: g(x)=1/(1-x)*sum{j>=1, x^(j^2)}=(theta_3(0,
               x)-1)/(1-x)/2 where theta_3 is a Jacobi theta function. - Hieronymus 
               Fischer (Hieronymus.Fischer(AT)gmx.de), May 26 2007
%p A000196 Digits := 100; A000196 := n->floor(evalf(sqrt(n)));
%t A000196 a[n_]:=IntegerPart[Sqrt[n]];lst={};Do[AppendTo[lst, a[n]], {n, 0, 6!}];
               lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 02 2008]
%o A000196 (MAGMA) [ Isqrt(n) : n in [0..100]];
%o A000196 (PARI) a(n)=floor(sqrt(n))
%o A000196 (PARI) a(n)=sqrtint(n)
%Y A000196 [A000267(n)/2]=A000196(n). Cf. A028391, A048766, A003056.
%Y A000196 Cf. A079051.
%Y A000196 Sequence in context: A068549 A132173 A023968 this_sequence A111850 A059396 
               A108602
%Y A000196 Adjacent sequences: A000193 A000194 A000195 this_sequence A000197 A000198 
               A000199
%K A000196 nonn,easy,nice
%O A000196 0,5
%A A000196 N. J. A. Sloane (njas(AT)research.att.com).