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A000037
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Numbers that are not squares.
(Formerly M0613 N0223)
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58
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2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Note the remarkable formula for the n-th term (see the FORMULA section)!
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
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
A010052(a(n)) = 0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010]
A173517(a(n)) = n; a(n)^2 = A030140(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 20 2010]
Special case of formula from Farhi for positive integers which are not r-th powers [Jonathan Vos Post, May 5, 2011].
Union of A007969 and A007970; A007968(a(n)) > 0. [Reinhard Zumkeller, Jun 18 2011]
Terms of even numbered rows in the triangle A199332. [Reinhard Zumkeller, Nov 23 2011]
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REFERENCES
| A. J. dos Reis and D. M. Silberger, "Generating nonpowers by formula", Mathematics Magazine 63 (1990), pp. 53-55.
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]
M. A. Nyblom, "Some curious sequences involving floor and ceiling functions", American Mathematical Monthly 109 (2002), pp. 559-564.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Bakir Farhi, An explicit formula generating the non-Fibonacci numbers, May 5, 2011.
S. R. Finch, Class number theory
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Continued Fraction
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FORMULA
| a(n) = n + [1/2 + sqrt(n)].
Another formula: a(n) = n + [ sqrt( n + [ sqrt n ] ) ].
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]
d(a(n))=even. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 20 2009]
a(n) = A000194(n) + n.
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EXAMPLE
| For example note that the squares 1, 4, 9, 16 are not included.
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]
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MAPLE
| A000037 := n->n+floor(1/2+sqrt(n));
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MATHEMATICA
| f[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[ f[n], {n, 71}] (from Robert G. Wilson v Sep 24 2004)
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]
With[{upto=100}, Complement[Range[upto], Range[Floor[Sqrt[upto]]]^2]] (* From Harvey P. Dale, Dec 02 2011 *)
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PROG
| (MAGMA) [n : n in [1..1000] | not IsSquare(n) ];
(MAGMA) at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; print at, n; end if; end for;
(PARI) a(n)=if(n<0, 0, n+(1+sqrtint(4*n))\2)
(Haskell)
a000037 n = n + a000196 (n + a000196 n)
-- Reinhard Zumkeller, Nov 23 2011
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CROSSREFS
| Cf. A007412, A000005, A000290, A059269.
Cf. A134986.
Cf. A087153, A172151. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010]
Cf. A000196.
Sequence in context: A072099 A046841 A164514 * A028761 A028809 A028785
Adjacent sequences: A000034 A000035 A000036 * A000038 A000039 A000040
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 30 2009
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