CarmichaelNumbers

The Carmichael numbers $561, 1105, 1729, …$ -- composite Fermat pseudoprimes to every coprime base -- as a lazy indexed collection.

CarmichaelNumbersthe composite

n

satisfying Korselt's criterion, an infinite indexed collection.

Domain: Collections

Details

A lazy indexed collection: $Count(CarmichaelNumbers) = +\infty$ (Alford–Granville–Pomerance, 1994), and $At$ unranks the $k$ -th by scanning forward, testing Korselt's criterion -- $n$ squarefree, and $(p-1) \mid (n-1)$ for every prime $p \mid n$ -- which is equivalent to passing the Fermat test $a^{n-1} \equiv 1 \pmod n$ for every $a$ coprime to $n$ .
OEIS A002997. $561 = 3 \cdot 11 \cdot 17$ is the smallest, found by Korselt in 1899 (four years before Carmichael's first published example).
Membership goes through Element: $Element(561, CarmichaelNumbers)$ is true, $Element(560, CarmichaelNumbers)$ is false.

Enumeration

1 more kept as data, checked against the oracles

Properties #

561 is the smallest Carmichael number

561∈\mathrm{CarmichaelNumbers}

Out

\mathrm{True}

560∈\mathrm{CarmichaelNumbers}

Out

\mathrm{False}

About

Examples	3
Crosswalk	OEIS mathlib4