A006446 - OEIS

%I M0548 #83 Sep 24 2022 05:48:21

%S 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,

%T 90,99,100,110,120,121,132,143,144,156,168,169,182,195,196,210,224,

%U 225,240,255,256,272,288,289,306,323,324,342,360,361,380,399,400,420

%N Numbers k such that floor(sqrt(k)) divides k.

%C 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

%C Union of A000290, A002378, and A005563. - _Fred Daniel Kline_, Feb 06 2016

%C The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1989). - _Amiram Eldar_, Jul 10 2020

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 21.

%D Jeffrey Shallit, personal communication.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Harry J. Smith, <a href="/A006446/b006446.txt">Table of n, a(n) for n = 1..1000</a>

%H Benoit Cloitre, <a href="http://dl.dropbox.com/u/46675017/divisibility_sequences.pdf">Some divisibility sequences</a>.

%H Curtis N. Cooper and Robert E. Kennedy, <a href="http://www.jstor.org/stable/2323194">Chebyshev's inequality and natural density</a>, Amer. Math. Monthly, Vol. 96, No. 2 (1989), pp. 118-124.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2319817">Problem E2491</a>, Amer. Math. Monthly, 82 (1975), 854-855.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).

%F 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

%F a(n) mod A000196(a(n)) = 0. - _Reinhard Zumkeller_, Apr 12 2013

%F a(n) = floor((n+1)/3)*(floor(n/3) + 1) +  floor((n+2)/3). - _Ridouane Oudra_, Nov 21 2020

%F 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

%F Sum_{n>=1} 1/a(n) = 7/4 + Pi^2/6. - _Amiram Eldar_, Sep 24 2022

%p A006446:=(-1-z-z**2+z**3)/(z**2+z+1)**2/(z-1)**3; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%p A006446:=n->`if`(type(n/floor(sqrt(n)), integer), n, NULL); seq(A006446(n), n=1..100); # _Wesley Ivan Hurt_, Feb 11 2014

%t Select[ Range[ 500 ], Mod[ #, Floor[ Sqrt[ # ]//N ] ]==0& ]

%o (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

%o (PARI) a(n)=my(k=n--\3+1);k*(k+n%3) \\ _Charles R Greathouse IV_, Jul 07 2011

%o (Haskell)

%o a006446 n = a006446_list !! (n-1)

%o a006446_list = filter (\x -> x `mod` a000196 x == 0) [1..]

%o -- _Reinhard Zumkeller_, Mar 31 2011

%Y Cf. A000196, A035106, A066377, A087811.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_