

A028391


a(n) = n  floor(sqrt(n)).


20



0, 0, 1, 2, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65
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OFFSET

0,4


COMMENTS

Number of nonsquares <= n.
Number of k <= n with an even number of divisors.  Benoit Cloitre, Sep 07 2002
Construct the pyramid
............a(0)
.......a(1).a(2).a(3)
..a(4).a(5).a(6).a(7).a(8).. etc.
Now circle all the primes and the result will be a pattern very similar to the famous Ulam spiral.  Sam Alexander, Nov 14 2003
The sequence floor(nn^(1/2)) gives the same numbers with a different offset.  Mohammad K. Azarian, R. J. Mathar and M. F. Hasler, Apr 30 2008
The number of nonzero values of floor (j^2/n) taken over 1 <= j <= n1.
a(n) = A173517(n) iff n is not a square.  Reinhard Zumkeller, Feb 20 2010
a(n)  a(n1) = 0 if n is a square, otherwise 1.  Robert Israel, Dec 30 2014


REFERENCES

B. Alspach, K. Heinrich and G. Liu, Orthogonal factorizations of graphs, pp. 1340 of Contemporary Design Theory, ed. J. H. Dinizt and D. R. Stinson, Wiley, 1992 (see Theorem 2.7).


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Dick Boland, Introduction to the Square Spine Spiral, 20002003 [broken link].


FORMULA

a(n) = ceiling(n  sqrt(n)), as follows from ceiling(x) = floor(x). [Corrected by M. F. Hasler, Feb 21 2010]
a(n) = 2*n  A028392(n).  Reinhard Zumkeller, Oct 28 2012
G.f.: (1+x)/(2*(1x)^2)  Theta3(0,x)/(2*(1x)) where Theta3 is a Jacobi theta function.  Robert Israel, Dec 30 2014


MAPLE

seq(n  floor(sqrt(n)), n = 0 .. 100); # Robert Israel, Dec 30 2014


MATHEMATICA

f[n_]:=nFloor[Sqrt[n]]; Table[f[n], {n, 0, 5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2010 *)


PROG

(Haskell)
a028391 n = n  a000196 n  Reinhard Zumkeller, Oct 28 2012
(PARI) a(n)=nsqrtint(n) \\ Charles R Greathouse IV, Jun 28 2013
(MAGMA) [nFloor(Sqrt(n)): n in [0..100]]; // Vincenzo Librandi Dec 31 2014


CROSSREFS

Cf. A056847, A000196, A135662A135665, A166373.
Sequence in context: A140859 A072586 A329548 * A038668 A251629 A279033
Adjacent sequences: A028388 A028389 A028390 * A028392 A028393 A028394


KEYWORD

nonn,easy,nice


AUTHOR

John Mellor (u15630(AT)snet.net)


EXTENSIONS

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, May 01 2008
Comment and crossreference added by Christopher Hunt Gribble, Oct 13 2009
Formula corrected by M. F. Hasler, Feb 21 2010
More terms from Vladimir Joseph Stephan Orlovsky, Mar 29 2010


STATUS

approved



