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Search: id:A000037
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| A000037 |
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Numbers that are not squares (note the remarkable formula for the n-th term).
(Formerly M0613 N0223)
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+0
33
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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; listen)
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OFFSET
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1,1
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COMMENT
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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
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REFERENCES
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M. A. Nyblom, Some curious sequences ..., Am. Math. Monthly 109 (#6, 200), 559-564.
A. J. dos Reis and D. M. Silberger, Generating nonpowers by formula, Math. Mag., 63 (1990), 53-55.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..10000
S. R. Finch, Class number theory
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1)
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2)
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FORMULA
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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]
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EXAMPLE
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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
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A000037 := n->n+floor(1/2+sqrt(n));
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MATHEMATICA
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f[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[ f[n], {n, 71}] (from Robert G. Wilson v Sep 24 2004)
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PROGRAM
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(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)
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CROSSREFS
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Cf. A007412, A000005, A000290, A059269.
Equals A000194(n) + n.
Cf. A134986.
Sequence in context: A028729 A072099 A046841 this_sequence A028761 A028809 A028785
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KEYWORD
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easy,nonn,nice,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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