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Search: id:A000196
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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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+0
83
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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; listen)
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OFFSET
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0,5
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COMMENT
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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 square-free 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
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REFERENCES
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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
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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
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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
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Digits := 100; A000196 := n->floor(evalf(sqrt(n)));
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MATHEMATICA
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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]
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PROGRAM
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(MAGMA) [ Isqrt(n) : n in [0..100]];
(PARI) a(n)=floor(sqrt(n))
(PARI) a(n)=sqrtint(n)
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CROSSREFS
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[A000267(n)/2]=A000196(n). Cf. A028391, A048766, A003056.
Cf. A079051.
Sequence in context: A068549 A132173 A023968 this_sequence A111850 A059396 A108602
Adjacent sequences: A000193 A000194 A000195 this_sequence A000197 A000198 A000199
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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