A125643 Squares and cubes (with repetition).

0, 0, 1, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681

Offset: 1

Author

Zak Seidov, Oct 19 2006

Keywords

• nonn

Comments

Repeating terms are sixth powers: 0,1,64,729,... (A001014).

For numbers not appearing as a difference between a square and an adjacent cube in this list, see A054504 and A081121.

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000.

Crossrefs

Cf. A002760 (squares and cubes (without repetitions)).

Cf. A001014, A054504, A081121, A087285, A087286, A088017.

Programs

Mathematica

m=1681;cm=Floor[m^(1/3)];sm=Floor[Sqrt[m]];s=Range[0,sm]^2;c=Range[0,cm]^3;Sort[Join[s,c]] (* James C. McMahon, Dec 20 2024 *)

Python

from math import isqrt
from sympy import integer_nthroot
def A125643(n):
    if n <= 4: return n-1>>1
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-2+x-integer_nthroot(x,3)[0]-isqrt(x)
    return bisection(f,n-2,n-2) # Chai Wah Wu, Oct 14 2024

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007