Are there factorions (numbers equal to the sum of the factorials of their digits) larger than 40,585?

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New entry, edited from material collected for the archive and not yet in a released version.

Problem

Are there factorions (numbers equal to the sum of the factorials of
their digits) larger than 40,585?

Factorions are numbers that are the sum of the factorial values for
each of their digits.  For example, 145 is a factorion because it can
be expressed as 145 = 1! + 4! + 5!.  The largest known factorion is
40,585 = 4! + 0! + 5! + 8! + 5!.

Some mathematicians believe that this is the largest possible
factorion.  (In fact, 145 and 40,585 are the only multi-digit
factorions known to humanity.)

Can you end the loneliness of the factorions?

-- Cliff Pickover <cliff@watson.ibm.com> (Cliff Puzzle 22: Factorions)

[ Editor: the first paragraph is the editor's summary. ]

Show solution

0, 1, 2, 145, and 40585 are the only factorions (assuming base 10, and
also assuming no extra leading 0's).

To see this, observe that an n digit factorion cannot exceed n*9!
Therefore a factorion can have at most 7 digits.  A quick computer
search of the 1-7 digit numbers shows the rest.

-- David Applegate <david@research.att.com>