AutomorphicNumbers

The automorphic numbers $1, 5, 6, 25, 76, 376, …$ : $n$ whose square ends in $n$ itself (base 10).

AutomorphicNumbersthe

n

with

n^2 \equiv n \pmod{10^{\mathrm{digits}(n)}}

, excluding the trivial 0, an infinite indexed collection.

Domain: Collections

Details

A lazy indexed collection: $Count(AutomorphicNumbers) = +\infty$ -- the two nontrivial 10-adic idempotents ( $…890625$ and $…109376$ ) extend to a new automorphic number of every digit length. OEIS A003226.
$At(AutomorphicNumbers, k)$ unranks from a table built by Hensel-lifting those idempotents digit by digit (not a search -- the terms thin out too fast, roughly 10x per step, for a scan to reach past the mid-teens); $At(AutomorphicNumbers, 5) = 76$ . Terms whose value exceeds $2^{53}-1$ answer $NaN$ , the same representable-range limit as NarcissisticNumbers.
Membership goes through Element: $Element(76, AutomorphicNumbers)$ is true ( $76^2=5776$ ), $Element(77, AutomorphicNumbers)$ is false.

Enumeration

1 more kept as data, checked against the oracles

About

Examples	1
Crosswalk	OEIS