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%I #52 Jul 28 2022 20:55:14
%S 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,
%T 22,23,24,25,26,27,28,29,30,30,31,32,33,34,35,36,37,38,39,40,41,42,42,
%U 43,44,45,46,47,48,49,50,51,52,53,54,55,56,56,57,58,59,60,61,62,63,64,65
%N a(n) = n - floor(sqrt(n)).
%C Number of nonsquares <= n.
%C Number of k <= n with an even number of divisors. - _Benoit Cloitre_, Sep 07 2002
%C Construct the pyramid
%C ............a(0)
%C .......a(1).a(2).a(3)
%C ..a(4).a(5).a(6).a(7).a(8).. etc.
%C Now circle all the primes and the result will be a pattern very similar to the famous Ulam spiral. - _Sam Alexander_, Nov 14 2003
%C The sequence floor(n-n^(1/2)) gives the same numbers with a different offset. - _Mohammad K. Azarian_, _R. J. Mathar_ and _M. F. Hasler_, Apr 30 2008
%C The number of nonzero values of floor (j^2/n) taken over 1 <= j <= n-1.
%C a(n) = A173517(n) iff n is not a square. - _Reinhard Zumkeller_, Feb 20 2010
%C a(n) - a(n-1) = 0 if n is a square, otherwise 1. - _Robert Israel_, Dec 30 2014
%D B. Alspach, K. Heinrich and G. Liu, Orthogonal factorizations of graphs, pp. 13-40 of Contemporary Design Theory, ed. J. H. Dinizt and D. R. Stinson, Wiley, 1992 (see Theorem 2.7).
%H Reinhard Zumkeller, <a href="/A028391/b028391.txt">Table of n, a(n) for n = 0..10000</a>
%H Dick Boland, <a href="http://imathination.net/SSS_first.htm">Introduction to the Square Spine Spiral</a>, 2000-2003 [broken link].
%F a(n) = ceiling(n - sqrt(n)), as follows from ceiling(-x) = -floor(x). [Corrected by _M. F. Hasler_, Feb 21 2010]
%F a(n) = 2*n - A028392(n). - _Reinhard Zumkeller_, Oct 28 2012
%F G.f.: (1+x)/(2*(1-x)^2) - Theta3(0,x)/(2*(1-x)) where Theta3 is a Jacobi theta function. - _Robert Israel_, Dec 30 2014
%p seq(n - floor(sqrt(n)), n = 0 .. 100); # _Robert Israel_, Dec 30 2014
%t f[n_]:=n-Floor[Sqrt[n]];Table[f[n],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 29 2010 *)
%o (Haskell)
%o a028391 n = n - a000196 n -- _Reinhard Zumkeller_, Oct 28 2012
%o (PARI) a(n)=n-sqrtint(n) \\ _Charles R Greathouse IV_, Jun 28 2013
%o (Magma) [n-Floor(Sqrt(n)): n in [0..100]]; // _Vincenzo Librandi_ Dec 31 2014
%o (Python)
%o from math import isqrt
%o def A028391(n): return n-isqrt(n) # _Chai Wah Wu_, Jul 28 2022
%Y Cf. A056847, A000196, A135662-A135665, A166373.
%K nonn,easy,nice
%O 0,4
%A John Mellor (u15630(AT)snet.net)
%E Edited by _N. J. A. Sloane_ at the suggestion of _R. J. Mathar_, May 01 2008
%E Comment and cross-reference added by _Christopher Hunt Gribble_, Oct 13 2009
%E Formula corrected by _M. F. Hasler_, Feb 21 2010
%E More terms from _Vladimir Joseph Stephan Orlovsky_, Mar 29 2010
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