|
| |
|
|
A000196
|
|
Integer part of square root of n. Or, number of squares <= n. Or, n appears 2n+1 times.
|
|
149
|
|
|
|
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;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
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, May 02 2002
Number of numbers k (<=n) with an odd number of divisors - Benoit Cloitre, 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]
|
|
|
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.
N. J. A. Sloane and Allan Wilks, On sequences of Recaman type, paper in preparation, 2006.
|
|
|
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!.
|
|
|
FORMULA
|
a(n) = a(n-1) + floor(n/(a(n-1)+1)^2), a(0) = 0. - Reinhard Zumkeller, 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, May 26 2007
a(n) = floor(sqrt(n)). - Arkadiusz Wesolowski, Jan 09 2013
|
|
|
MAPLE
|
Digits := 100; A000196 := n->floor(evalf(sqrt(n)));
|
|
|
MATHEMATICA
|
Table[n, {n, 0, 20}, {2n+1}]//Flatten (* Zak Seidov Mar 19 2011 *)
IntegerPart[Sqrt[Range[0, 110]]] (* Harvey P. Dale, May 23 2012 *)
|
|
|
PROG
|
(MAGMA) [ Isqrt(n) : n in [0..100]];
(PARI) a(n)=floor(sqrt(n))
(PARI) a(n)=sqrtint(n)
(Haskell)
import Data.Bits (shiftL, shiftR)
a000196 :: Integer -> Integer
a000196 0 = 0
a000196 n = newton n (findx0 n 1) where
-- find x0 == 2^(a+1), such that 4^a <= n < 4^(a+1).
findx0 0 b = b
findx0 a b = findx0 (a `shiftR` 2) (b `shiftL` 1)
newton n x = if x' < x then newton n x' else x
where x' = (x + n `div` x) `div` 2
a000196_list = concat $ zipWith replicate [1, 3..] [0..]
-- Reinhard Zumkeller, Apr 12 2012, Oct 23 2010
|
|
|
CROSSREFS
|
[A000267(n)/2]=A000196(n). Cf. A000290, A028391, A048766, A074704, A003056, A079051.
Sequence in context: A023968 A204166 A178786 * A111850 A059396 A108602
Adjacent sequences: A000193 A000194 A000195 * A000197 A000198 A000199
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
