PowersOfTwo

The powers of two $1, 2, 4, 8, …$ — subsets of an $n$-set; row-sums of Pascal's triangle.

OEISA000079↗

PowersOfTwo$2^{n-1}$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(PowersOfTwo) = +\infty$, and $At(PowersOfTwo, k) = 2^{k-1}$ unranks in closed form, exact for every $k$ -- $At(PowersOfTwo, 60)$ is exact past $2^{53}$, not a rounded double.
• OEIS A000079.
• Membership goes through Element by the bit trick $x \mathbin{\&} (x-1) = 0$ -- $Element(64, PowersOfTwo)$ is true, $Element(48, PowersOfTwo)$ is false.
• $Count(Subsets(n)) = At(PowersOfTwo, n+1)$; see Subsets.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

2 ^ _r

rank(_x)

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

valid(_x)

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

Enumeration

Examples

1 more kept as data, checked against the oracles

Properties #

live ↗

In$64∈\mathrm{PowersOfTwo}$Out$\mathrm{True}$

live ↗

In$48∈\mathrm{PowersOfTwo}$Out$\mathrm{False}$

About

Examples	3
Crosswalk	OEIS

See also: Subsets, FactorialNumbers, Count, At, Element