%I A000037 M0613 N0223
%S A000037 2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,
%T A000037 29,30,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,50,51,
%U A000037 52,53,54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,
               76,77,78,79,80,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,
               99
%N A000037 Numbers that are not squares (note the remarkable formula for the n-th 
               term).
%C A000037 These are the natural numbers with an even number of divisors. The number 
               of divisors is odd for the complementary sequence, the squares (sequence 
               A000290) and the numbers for which the number of divisors is divisible 
               by 3 is sequence A059269. - Ola Veshta (olaveshta(AT)my-deja.com), 
               Apr 04 2001
%C A000037 Also, a(n) = largest integer m not equal to n such that n = (floor(n^2/
               m) + m)/2. - Alexander R. Povolotsky (pevnev(AT)juno.com), Feb 10 
               2008
%D A000037 A. J. dos Reis and D. M. Silberger, "Generating nonpowers by formula", 
               Mathematics Magazine 63 (1990), pp. 53-55.
%D A000037 M. A. Nyblom, "Some curious sequences involving floor and ceiling functions", 
               American Mathematical Monthly 109 (2002), pp. 559-564.
%D A000037 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000037 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000037 J. Lambek and L. Moser, "Inverse and complementary sequences of natural 
               numbers", The American Mathematical Monthly, Vol. 61, No. 7 (1954), 
               454-458, doi 10.2307/2308078, see example 4 (includes the formula). 
               [From Nicolas Normand (Nicolas.Normand(AT)polytech.univ-nantes.fr), 
               Nov 24 2009]
%H A000037 N. J. A. Sloane, <a href="b000037.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A000037 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Class number theory</
               a>
%H A000037 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareNumber.html">Square Number</a>
%H A000037 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ContinuedFraction.html">Continued Fraction</a>
%F A000037 a(n) = n + [1/2 + sqrt(n)].
%F A000037 Another formula: a(n) = n + [ sqrt( n + [ sqrt n ] ) ].
%F A000037 a(n) = A000194(n) + n = floor(1/2 *(1 + sqrt(4*n-3)))+ n. [From Jaroslav 
               Krizek (jaroslav.krizek(AT)atlas.cz), Jun 14 2009]
%F A000037 d(a(n))=even. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 
               20 2009]
%e A000037 For example note that the squares 1, 4, 9, 16 are not included.
%e A000037 a(A002061(n)) = a(n^2-n+1) = A002522(n) = n^2 + 1. A002061(n) = central 
               polygonal numbers (n^2-n+1). A002522(n) = numbers of the form n^2 
               + 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 21 
               2009]
%p A000037 A000037 := n->n+floor(1/2+sqrt(n));
%t A000037 f[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[ f[n], {n, 71}] 
               (from Robert G. Wilson v Sep 24 2004)
%t A000037 f[n_]:=Round[Sqrt[n]]; lst={};Do[AppendTo[lst,n+f[n]],{n,0,5!}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]
%o A000037 (MAGMA) [n : n in [1..1000] | not IsSquare(n) ];
%o A000037 (MAGMA) at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; 
               print at, n; end if; end for;
%o A000037 (PARI) a(n)=if(n<0,0,n+(1+sqrtint(4*n))\2)
%Y A000037 Cf. A007412, A000005, A000290, A059269.
%Y A000037 Equals A000194(n) + n.
%Y A000037 Cf. A134986.
%Y A000037 Sequence in context: A072099 A046841 A164514 this_sequence A028761 A028809 
               A028785
%Y A000037 Adjacent sequences: A000034 A000035 A000036 this_sequence A000038 A000039 
               A000040
%K A000037 easy,nonn,nice,new
%O A000037 1,1
%A A000037 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A000037 Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 
               30 2009