NarcissisticNumbers

The Armstrong (narcissistic) numbers $1, …, 9, 153, 370, …$: $n$ equal to the sum of its own digits, each raised to the digit count -- the one PROVEN finite family here.

OEISA005188↗

NarcissisticNumbersthe base-10 narcissistic numbers, excluding the trivial 0: a finite indexed collection of exactly 88.

Domain: Collections

Details

• A lazy indexed collection, but $Count(NarcissisticNumbers) = 88$, not $+\infty$: Diamond & Kellner's digit-length bound proves base 10 has no more, and the largest is the 39-digit $115132219018763992565095597973971522401$. OEIS A005188 (which also lists the trivial $0 = 0^1$; this family starts at 1).
• $At(NarcissisticNumbers, k)$ unranks from a table verified against every OEIS term -- $At(NarcissisticNumbers, 10) = 153$. Terms past the 43rd exceed $2^{53}-1$ (IEEE-754 double precision) and come back as exact bigint instead, so every one of the 88 -- including the 39-digit largest -- is exact, never $NaN$.
• Membership goes through Element: $Element(153, NarcissisticNumbers)$ is true ($1^3+5^3+3^3=153$), $Element(154, NarcissisticNumbers)$ is false.

Enumeration

Examples

1 more kept as data, checked against the oracles

Scope #

The 88th and largest term, a 39-digit exact integer well past $2^{53}-1$

live ↗

In$\mathrm{NarcissisticNumbers}[88]$Out$115\,132\,219\,018\,763\,992\,565\,095\,597\,973\,971\,522\,401$

Properties #

Narcissistic (Armstrong) numbers are proven finite: exactly 88 exist.

live ↗

In$\mathrm{Count}(\mathrm{NarcissisticNumbers})$Out$88$

About

Examples	3
Crosswalk	OEIS

See also: Count, At, Element, KaprekarNumbers