A006446 Numbers k such that floor(sqrt(k)) divides k.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

References

• 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).

Links

• 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).

Crossrefs

Cf. A000196, A035106, A066377, A087811.

Programs

Haskell

a006446 n = a006446_list !! (n-1)
a006446_list = filter (\x -> x `mod` a000196 x == 0) [1..]
-- Reinhard Zumkeller, Mar 31 2011

Maple

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

Mathematica

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

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

Formula

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

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