Need help with sequences

[Max A.]

On 12/27/06, Tanya Khovanova <tanyakh at tanyakhovanova.com> wrote:

> Number 1:
> 4624, 595968 - Numbers n such that the sum of the 4 to the powers of its digits equals n.
> Comment: If you use 2 instead of 4, such numbers do not exist. If you use 3, the only number is 12.
> Example: 4^4 + 4^6 + 4^2 + 4^4 = 256 + 4096 + 16 + 256 = 4624.
> Math: Select[Range[10000000], Plus @@ (4^IntegerDigits[#]) == # &]

This is a finite sequence, and you have found all its terms.

Indeed, if the number consists of n decimal digits then
1) the sum of the 4th powers of its digits is at most n*9^4;
2) the number itself is at least 10^(n-1).
But it is easy to see that 10^(n-1) > n*9^4 for n>=6. Therefore, all
numbers in this sequence have at most 5 decimal digits.

Max