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A006446
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Numbers k such that floor(sqrt(k)) divides k.
(Formerly M0548)
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25
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1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 20, 24, 25, 30, 35, 36, 42, 48, 49, 56, 63, 64, 72, 80, 81, 90, 99, 100, 110, 120, 121, 132, 143, 144, 156, 168, 169, 182, 195, 196, 210, 224, 225, 240, 255, 256, 272, 288, 289, 306, 323, 324, 342, 360, 361, 380, 399, 400, 420
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Numbers of the form k^2, k*(k+1), or k*(k+2). Nonsquare elements of this sequence are given by A035106. - Max Alekseyev, Nov 27 2006
Union of A000290, A002378, and A005563. - Fred Daniel Kline, Feb 06 2016
The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1989). - Amiram Eldar, Jul 10 2020
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 21.
Jeffrey Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 1..1000
Benoit Cloitre, Some divisibility sequences
Curtis N. Cooper and Robert E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly, Vol. 96, No. 2 (1989), pp. 118-124.
S. W. Golomb, Problem E2491, Amer. Math. Monthly, 82 (1975), 854-855.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).
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FORMULA
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For k>=1 a(3*k-2) = k^2, a(3*k-1) = k*(k+1) and a(3*k) = k*(k+2). - Benoit Cloitre, Jan 14 2012
a(n) mod A000196(a(n)) = 0. - Reinhard Zumkeller, Apr 12 2013
a(n) = floor((n+1)/3)*(floor(n/3) + 1) + floor((n+2)/3). - Ridouane Oudra, Nov 21 2020
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7. - Chai Wah Wu, Apr 05 2021
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MAPLE
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A006446:=(-1-z-z**2+z**3)/(z**2+z+1)**2/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation
A006446:=n->`if`(type(n/floor(sqrt(n)), integer), n, NULL); seq(A006446(n), n=1..100); # Wesley Ivan Hurt, Feb 11 2014
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MATHEMATICA
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Select[ Range[ 500 ], Mod[ #, Floor[ Sqrt[ # ]//N ] ]==0& ]
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PROG
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(PARI) { n=0; for (m=1, 10^9, if (m%floor(sqrt(m)) == 0, write("b006446.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 12 2010
(PARI) a(n)=my(k=n--\3+1); k*(k+n%3) \\ Charles R Greathouse IV, Jul 07 2011
(Haskell)
a006446 n = a006446_list !! (n-1)
a006446_list = filter (\x -> x `mod` a000196 x == 0) [1..]
-- Reinhard Zumkeller, Mar 31 2011
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CROSSREFS
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Cf. A000196, A035106, A066377, A087811.
Sequence in context: A231404 A316860 A097273 * A261342 A002348 A019469
Adjacent sequences: A006443 A006444 A006445 * A006447 A006448 A006449
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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