Perfect digit to digit invariant

A perfect digit-to-digit invariant (PDDI) (also known as a Munchausen number) is a natural number that is equal to the sum of its digits each raised to a power equal to the digit.

n = d k d k + d k − 1 d k − 1 + ⋯ + d 2 d 2 + d 1 d 1 .

0 and 1 are PDDIs in any base (using the convention that 0⁰ = 0). Apart from 0 and 1 there are only two other PDDIs in the decimal system, 3435 and 438579088 (sequence A046253 in the OEIS). Note that the second of these is only a PDDI under the convention that 0⁰ = 0, but this is standard usage in this area.

3 3 + 4 4 + 3 3 + 5 5 = 27 + 256 + 27 + 3125 = 3435 4 4 + 3 3 + 8 8 + 5 5 + 7 7 + 9 9 + 0 0 + 8 8 + 8 8

More generally, there are finitely many PDDIs in any base. This can be proved as follows:

Let b be a base. Every PDDI n in base b is equal to the sum of its digits each raised to a power equal to the digit. This sum is less than or equal to a ( b − 1 ) b − 1 , where a is the number of digits in n , because b − 1 is the largest possible digit in base b . Thus, a ( b − 1 ) b − 1 ≥ n ≥ b a − 1 . The expression a ( b − 1 ) b − 1 increases linearly with respect to a , whereas the expression b a − 1 increases exponentially with respect to a . So there is some k > 0 such that ∀ a ≥ k , a ( b − 1 ) b − 1 < b a − 1 . There are finitely many natural numbers n with fewer than k digits, so there are finitely many natural numbers n satisfying the first inequality. Thus, there are only finitely many PDDIs in base b .

In all bases 1 is a PDDI.
In base 3 there are 2 PDDI's, namely 12 and 22. (5 and 8 in decimals)
In base 4 there are 2 PDDI's, namely 131 and 313. (29 and 55 in decimals)
In base 6 there are 2 PDDI's, namely 22352 and 23452. (3164 and 3416 in decimals)
In base 7 there is 1 PDDI's, namely 13454. (3665 in decimals)
In base 9 there are 3 PDDI's, namely 31, 156262 and 1656547. (28, 96446 and 923362 in decimals)

When the convention 0 0 = 0 is used the following numbers are also PDDI's

In all bases 0 is a PDDI.
In base 4 there is one additional PDDI, namely 130. (28 in decimal)
In base 5 there are 2 PDDI's, namely 103 and 2024. (28 and 264 in decimal)
In base 8 there are 2 PDDI's, namely 400 and 401. (256 and 257 in decimal)
In base 9 there are 3 additional PDDI's, namely 30, 1647063 and 34664084. (27, 917139 and 16871323 in decimal)