40,585 has a property closely related to, but genuinely distinct from, the digit-power self-construction seen in Münchhausen numbers like 3,435: instead of raising each digit to a power equal to itself, a factorion sums the *factorial* of each of its own digits and reproduces the original number exactly. 40,585's digits are 4, 0, 5, 8, and 5, so the calculation is 4! + 0! + 5! + 8! + 5! = 24 + 1 + 120 + 40,320 + 120 = 40,585 — exactly matching the number those digits came from.
A Provably Closed List of Four
Factorions are genuinely, provably rare in base 10 — there are only four of them in total, and this isn't a matter of none having been found yet the way it is with odd perfect numbers; it's a completely settled, proven fact. The four base-10 factorions are 1 (trivially, 1!=1), 2 (trivially, 2!=2), 145 (=1!+4!+5! = 1+24+120=145), and 40,585, with no others existing at any size — a boundedness that follows from the same kind of counting argument used to prove Münchhausen numbers are similarly limited: because 9! = 362,880, a number with d digits can have a digit-factorial sum of at most 362,880×d, but the number itself grows roughly as 10^(d−1), meaning the digit-factorial sum is eventually and permanently outpaced by the number of digits itself once d grows large enough — putting a hard, provable ceiling on how large a factorion can possibly be, rather than leaving the question open to further search the way some other digit-based curiosities remain.
145: the Smaller Companion Example
145 is worth knowing alongside 40,585 as the other genuinely nontrivial example in this small family, and it's a useful, smaller number to verify the concept by hand before tackling 40,585's larger arithmetic: 1! + 4! + 5! = 1 + 24 + 120 = 145, exactly. Both 145 and 40,585 share the specific quality that makes factorions genuinely interesting rather than merely curious — like Münchhausen numbers, they're reconstructed entirely and exactly from a specific operation applied to their own digits, with no external reference number involved at all.
As a plain integer, 40,585 factors as 5 × 8,117, where 8,117 is itself prime — giving 40,585 exactly four divisors (1, 5, 8117, 40585), a comparatively sparse divisor list for a five-digit number, directly because one of its only two prime factors is a fairly large prime in its own right rather than a smaller, more composite-friendly building block. This ordinary factorization has no direct bearing on its factorion status, the same way 3,435's ordinary factorization has no bearing on its Münchhausen property — both numbers are famous for a digit-based construction that sits entirely separate from their prime-factorization structure.
Why Both Families Are Finite
Both 40,585 and 3,435, despite belonging to genuinely different digit-based number families (factorions and Münchhausen numbers, respectively), share a structural reason for their extreme rarity worth stating side by side: both properties compare a number's own magnitude against a digit-based sum that grows only in proportion to the number's *digit count*, not its actual size, meaning the digit-based sum is eventually and permanently overtaken by the raw magnitude of larger numbers, capping how large a member of either family can possibly be. This shared underlying reasoning is a genuinely useful, transferable insight for evaluating any newly proposed "self-referential digit property" claim: if the defining sum grows only linearly with digit count while the number itself grows exponentially, the family is guaranteed to be finite, and often quite small.
145's own discovery predates the formal "factorion" terminology by a considerable margin, having circulated informally in recreational-mathematics circles for many decades as a standalone curiosity before anyone had systematically searched for and proven the completeness of the full base-10 family it belongs to — a pattern common to several entries on this site, where a specific number's charm was noticed and enjoyed long before mathematicians formally proved the boundaries of the broader category it turned out to belong to. 40,585's own place as the largest member of that same small, now-completed family gives the two numbers a genuinely tidy shared story, discovered separately but bound together by the same underlying digit-factorial rule, one small and easily hand-checked, the other considerably larger yet governed by the exact same self-referential logic throughout.