PlanetMath: factorion

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factorion

(Definition)

Given a base integer

$\displaystyle n = \sum_{i = 1}^k d_ib^{i - 1}$

where

is the least significant digit and

is the most significant, if it is also the case that

$\displaystyle n = \sum_{i = 1}^k d_i!$

then

is a factorion. In other words, the sum of the factorials of the digits in a standard positional integer base

(such as base 10) gives the same number as multiplying the digits by the appropriate power of that base. With the exception of 1, the factorial base representation of a factorion is always different from that in the integer base. Obviously, all numbers are factorions in factorial base.

1 is a factorion in any integer base. 2 is a factorion in all integer bases except binary. In base 10, there are only four factorions: 1, 2, 145 and 40585. For example, $4 \times 10^4 + 0 \times 10^3 + 5 \times 10^2 + 8 \times 10^1 + 5 \times 10^0 = 4! + 0! + 5! + 8! + 5! = 40585$ . (The factorial base representation of 40585 is 10021001).

Bibliography

1: D. Wells, The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 125

"factorion" is owned by Mravinci.

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Cross-references: binary, bases, representation, factorial base, number, digits, factorials, sum, least significant digit, integer, base
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This is version 3 of factorion, born on 2008-01-08, modified 2008-01-11.
Object id is 10180, canonical name is Factorion.
Accessed 248 times total.

Classification:

AMS MSC:	05A10 (Combinatorics :: Enumerative combinatorics :: Factorials, binomial coefficients, combinatorial functions)
	11B65 (Number theory :: Sequences and sets :: Binomial coefficients; factorials; $q$-identities)
	11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems)

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