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A000196
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Integer part of square root of n. Or, number of squares <= n. Or, n appears 2n+1 times.
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121
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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, 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Also the integer part of the geometric mean of the divisors of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 19 2001
a(n)=Card(k, 0<k<=n such that k is relatively prime to core(k)) where core(x) is the squarefree part of x. - Benoit Cloitre (benoit7848c(AT)orange.fr), May 02 2002
Number of numbers k (<=n) with an odd number of divisors - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 07 2002
Also, for n > 0, the number of digits when writing n in base where place values are squares, cf. A007961; A190321(n) <= a(n). [Reinhard Zumkeller, May 08 2011]
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REFERENCES
| T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 23.
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.
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
N. J. A. Sloane and A. R. Wilks, On sequences of Recaman type, paper in preparation, 2006.
F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
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LINKS
| Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000
K. Atanassov, On Some of Smarandache's Problems
H. Bottomley, Illustration of A000196, A048760, A053186
F. Smarandache, Only Problems, Not Solutions!.
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FORMULA
| 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
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
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MAPLE
| Digits := 100; A000196 := n->floor(evalf(sqrt(n)));
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MATHEMATICA
| 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]
Table[n, {n, 0, 20}, {2n+1}]//Flatten (* Zak Seidov Mar 19 2011 *)
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PROG
| (MAGMA) [ Isqrt(n) : n in [0..100]];
(PARI) a(n)=floor(sqrt(n))
(PARI) a(n)=sqrtint(n)
(Haskell)
a000196 0 = 0
a000196 n = if (r + 1) ^ 2 > n then r else r + 1
where r = 2 * (a000196 $ n `div` 4)
-- Cf. L. C. Paulson, ML for the Working Programmer, CUP 1996, page 52, 2.16
-- Reinhard Zumkeller, Oct 23 2010
-- Simpler variant:
a000196 n = length $ takeWhile (<= n) $ tail a000290_list
- Reinhard Zumkeller, Mar 18 2011, May 08 2011
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CROSSREFS
| [A000267(n)/2]=A000196(n). Cf. A000290, A028391, A048766, A074704, A003056.
Cf. A079051.
Sequence in context: A023968 A204166 A178786 * A111850 A059396 A108602
Adjacent sequences: A000193 A000194 A000195 * A000197 A000198 A000199
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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