CubeNumbers

The perfect cubes $1, 8, 27, 64, …$ — $n^3$.

OEISA000578↗

CubeNumbers$n^3$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(CubeNumbers) = +\infty$, and $At(CubeNumbers, k) = k^3$ unranks in closed form -- $At(CubeNumbers, 5) = 125$.
• OEIS A000578.
• Membership goes through Element by bisecting the monotone formula $n \mapsto n^3$ for its exact integer cube root -- $Element(125, CubeNumbers)$ is true, $Element(126, CubeNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

(_r + 1) ^ 3

rank(_x)

Apply(
       (sq_at: integer) => sq_at if (sq_at + 1) ^ 3 == _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) ^ 3 <= _x else sq_at,
                      sq_at,
                      sq_b,
                    ), 0, Range(Floor(Log(_x, 2)) + 1, 0, -1)),
     ) if True else -1

valid(_x)

Apply(
       (sq_at: integer) => (sq_at + 1) ^ 3 == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) ^ 3 <= _x else sq_at,
                      sq_at,
                      sq_b,
                    ), 0, Range(Floor(Log(_x, 2)) + 1, 0, -1)),
     ) if True && _x >= 1 else False

Enumeration

Examples

1 more kept as data, checked against the oracles

About

Examples	1
Crosswalk	OEIS

See also: SquareNumbers, TetrahedralNumbers, Count, At, Element