A377025 Squares and cubes that are not 6th powers.

4, 8, 9, 16, 25, 27, 36, 49, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025

Offset: 1

Author

Chai Wah Wu, Oct 13 2024

Keywords

• nonn
• easy

Comments

Squares and cubes that cannot be written as both a square and a cube.

{A002760}\{A001014}.

A125643 minus the repeated terms.

Crossrefs

Cf. A002760, A125643, A001014.

Programs

Mathematica

lim=2025;Select[Union[Range[Floor[lim^(1/2)]]^2,Range[Floor[lim^(1/3)]]^3],!IntegerQ[#^(1/6)]&] (* James C. McMahon, Oct 16 2024 *)

Python

from math import isqrt
from sympy import integer_nthroot
def A377025(n):
    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+x+(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,3)[0]-isqrt(x)
    return bisection(f,n,n)