A048760 Largest square <= n.

0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Author

Charles T. Le (charlestle(AT)yahoo.com)

Keywords

• nonn
• easy
• look

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Krassimir T. Atanassov, On Some of Smarandache's Problems, 1999.
• Henry Bottomley, Illustration of A000196, A048760, A053186.
• Jose Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), pp. 202-204.
• Valentina V. Radeva and Krassimir T. Atanassov, On the 40-th and 41-st Smarandache's problems, Notes on Number Theory and Discrete Mathematics, Vol. 4, No. 3 (1998), pp. 101-104.
• Florentin Smarandache, Only Problems, Not Solutions!, 1993.
• Michael Somos, Sequences used for indexing triangular or square arrays.

Crossrefs

Cf. A000196, A000290, A048761, A345202.

Programs

Haskell

a048760 = (^ 2) . a000196  -- Reinhard Zumkeller, Feb 12 2012

Maple

A048760 := proc(n)
    floor(sqrt(n)) ;
    %^2 ;
end proc: # R. J. Mathar, May 19 2016

Mathematica

Array[Floor[Sqrt[#]]^2&,80,0] (* Harvey P. Dale, Mar 30 2012 *)
Table[PadRight[{},2n+1,n^2],{n,0,10}]//Flatten (* Harvey P. Dale, Feb 28 2025 *)

PARI

a(n) = sqrtint(n)^2; \\ Michel Marcus, Jun 06 2015

Formula

a(n) = floor(n^(1/2))^2 = A000290(A000196(n)). - Reinhard Zumkeller, Feb 12 2012, Sep 03 2002

n^2 repeated (2n+1) times, n=0,1,... - Zak Seidov, Oct 25 2008

Sum_{n>=1} (1/a(n) - 1/n) = gamma + zeta(2) (= A345202). - Amiram Eldar, Jun 12 2021

Sum_{n>=1} 1/a(n)^2 = 2*zeta(3) + Pi^4/90. - Amiram Eldar, Aug 15 2022