A048761 Smallest square greater than or equal to n.

0, 1, 4, 4, 4, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 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, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

From M. F. Hasler, Oct 05 2009: (Start)

For each k > 0, the term k^2 is listed 2k - 1 times.

a(n+1) is the least square greater than n. (End)

References

• Krassimir Atanassov, On the 40th and 41st Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4, No. 3 (1998), 101-104.
• J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 202-204.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Krassimir Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 27-32.
• Florentin Smarandache, Only Problems, Not Solutions!.

Crossrefs

Cf. A000196, A010052, A048760, A165775.

Programs

Haskell

a048761 n = (a000196 n + 1 - a010052 n) ^ 2
a048761_list = 0 : concat (f 1 1) where
   f u v = (take v $ repeat u) : f (u + v + 2) (v + 2)
-- Reinhard Zumkeller, Mar 16 2014

Magma

[Ceiling(Sqrt(n))^2: n in [0..80]]; // Vincenzo Librandi, Jun 21 2015

Maple

A048761 := proc(n)
        ceil(sqrt(n)) ;
        %^2 ;
end proc: # R. J. Mathar, Sep 26 2011

Mathematica

(Ceiling[Sqrt[Range[0, 99]]])^2 (* Alonso del Arte, Jun 21 2015 *)

PARI

A048761(n)=if(n,(sqrtint(n-1)+1)^2,0) \\ M. F. Hasler, Oct 05 2009

Formula

a(n) = (A000196(n) + 1 - A010052(n))^2. - Reinhard Zumkeller, Mar 16 2014

a(n) = (ceiling(sqrt(n)))^2. - Alonso del Arte, Jun 21 2015

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

Extensions

Missing a(49) = 49 inserted by Reinhard Zumkeller, Mar 16 2014