SquareFreeNumbers

The squarefree numbers $1, 2, 3, 5, 6, …$ -- integers divisible by no perfect square $> 1$ -- as a lazy indexed collection.

OEISA005117↗mathlib4Squarefree↗

SquareFreeNumbersthe $n$ with every prime factor's exponent $\le 1$, an infinite indexed collection.

Domain: Collections

Details

• A lazy indexed collection: $Count(SquareFreeNumbers) = +\infty$ (density $6/\pi^2$), and $At$ unranks the $k$-th by scanning forward, testing each candidate's factorisation.
• OEIS A005117. Exactly $[[KFreeIntegers]](2)$ -- the two families are unranked identically.
• Membership goes through Element: $Element(10, SquareFreeNumbers)$ is true ($10 = 2 \cdot 5$), $Element(12, SquareFreeNumbers)$ is false ($12 = 2^2 \cdot 3$).

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Apply(
       (sn_f: integer) => sn_f if sn_f > 0 else `NaN`,
       Fold(Function(
                      sn_s if sn_s > 0 else (sn_k if sn_s + _r == 0 else sn_s -
                                                                         1) if Fold(
                                                                                    Function(ok_sf &&
                                                                                             sn_k %
                                                                                             (sf * sf) !=
                                                                                             0, ok_sf, sf),
                                                                                    True,
                                                                                    Range(2, Floor(Sqrt(sn_k)), 1),
                                                                                  ) else sn_s,
                      sn_s,
                      sn_k,
                    ), 0, Range(1, 2 * (_r + 1), 1)),
     )

rank(_x)

Fold(Function(
               sn_c + 1 if Fold(Function(
                                          ok_sf && sn_k % (sf * sf) != 0,
                                          ok_sf,
                                          sf,
                                        ), True, Range(2, Floor(Sqrt(sn_k)), 1)) else sn_c,
               sn_c,
               sn_k,
             ), 0, Range(1, _x - 1, 1)) if (Fold(
                                                 Function(
                                                           ok_sf && _x % (sf *
                                                                          sf) != 0,
                                                           ok_sf,
                                                           sf,
                                                         ),
                                                 True,
                                                 Range(2, Floor(Sqrt(_x)), 1),
                                               ) if _x >= 1 else False) else -1

valid(_x)

Fold(
      Function(ok_sf && _x % (sf * sf) != 0, ok_sf, sf),
      True,
      Range(2, Floor(Sqrt(_x)), 1),
    ) if _x >= 1 else False

Enumeration

Examples

1 more kept as data, checked against the oracles

About

Examples	1
Crosswalk	OEISmathlib4

See also: Count, At, Element, KFreeIntegers, PowerfulNumbers