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

1,5

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.

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A025475 A246547 A195942 * A002760 A377025 A355062

Adjacent sequences: A125640 A125641 A125642 * A125644 A125645 A125646

Keyword

nonn

Author

Zak Seidov, Oct 19 2006

Extensions

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

Status

approved