SquareNumbers

The perfect squares $1, 4, 9, 16, …$ as a lazy indexed collection, unranked by position.

OEISA000290↗Wolfram LanguagePolygonalNumber↗mathlib4IsSquare↗

SquareNumbersthe squares $k^2$ for $k = 1, 2, 3, …$, an infinite indexed collection.

Domain: Collections

Details

• A lazy indexed collection: $Count(SquareNumbers) = +\infty$, and $At(SquareNumbers, k) = k^2$ unranks in closed form -- $At(SquareNumbers, 5) = 25$.
• OEIS A000290.
• Membership goes through Element: $Element(16, SquareNumbers)$ is true, $Element(15, SquareNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

(_r + 1) * (_r + 1)

rank(_x)

Apply(
       (sq_at: integer) => sq_at if (sq_at + 1) * (sq_at + 1) == _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) *
                                          (sq_at + 2 ^ sq_b + 1) <= _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) * (sq_at + 1) == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) *
                                          (sq_at + 2 ^ sq_b + 1) <= _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

Basic #

The first five squares

live ↗

In$\mathrm{Take}(\mathrm{SquareNumbers}, 5)$Out$\bigl\lbrack1, 4, 9, 16, 25\bigr\rbrack$

About

Examples	1
Crosswalk	OEISWolfram Languagemathlib4

See also: Count, At, Element, PrimeNumbers, AbundantNumbers