A201053 Nearest cube.

0, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Author

Reinhard Zumkeller, Nov 28 2011

Keywords

• nonn
• look

Comments

a(n) = if n-A048763(n) < A048762(n)-n then A048762(n) else A048763(n);

apart from 0, k^3 occurs 3*n^2+1 times, cf. A056107.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A061023, A074989, A053187 (nearest square), A000578.

Cf. A048762, A048763, A056107.

Programs

Haskell

a201053 n = a201053_list !! n
a201053_list = 0 : concatMap (\x -> replicate (a056107 x) (x ^ 3)) [1..]

Maple

seq(k^3 $ (3*k^2+1), k=0..10); # Robert Israel, Jan 03 2017

Mathematica

Module[{nn=70,c},c=Range[0,Ceiling[Surd[nn,3]]]^3;Flatten[Array[ Nearest[ c,#]&,nn,0]]] (* Harvey P. Dale, May 27 2014 *)

Python

from sympy import integer_nthroot
def A201053(n):
    a = integer_nthroot(n,3)[0]
    return a**3 if 2*n < a**3+(a+1)**3 else (a+1)**3 # Chai Wah Wu, Mar 31 2021

Formula

G.f.: (1-x)^(-1)*Sum_{k>=0} (3*k^2+3*k+1)*x^((k+1)*(k^2+k/2+1)). - Robert Israel, Jan 03 2017

Sum_{n>=1} 1/a(n)^2 = Pi^4/30 + Pi^6/945. - Amiram Eldar, Aug 15 2022