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A028391
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a(n) = n - floor(sqrt(n)).
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21
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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
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0,4
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COMMENTS
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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 number of nonzero values of floor (j^2/n) taken over 1 <= j <= n-1.
a(n) - a(n-1) = 0 if n is a square, otherwise 1. - Robert Israel, Dec 30 2014
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) = ceiling(n - sqrt(n)), as follows from ceiling(-x) = -floor(x). [Corrected by M. F. Hasler, Feb 21 2010]
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
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MAPLE
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seq(n - floor(sqrt(n)), n = 0 .. 100); # Robert Israel, Dec 30 2014
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MATHEMATICA
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PROG
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(Haskell)
(Python)
from math import isqrt
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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John Mellor (u15630(AT)snet.net)
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EXTENSIONS
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STATUS
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approved
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