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### Best Answer Sid Kemp says

I believe that there is no whole number power of two who's digits add up to 30, I checked everything from 1 to 33, and there wasn't. At that point, the total of the digits was 62. Of course, sometimes the total goes down, as the digits go higher. But everything from 2^26 had a digit sum greater than 30. From 2^24 on up, the total number of digits is 11 or more, so all would have to be very low digits to create a total of 30 (average digit would be below 3). Assuming a pseudorandom result of digits (base 10) for powers of 2, it is very unlikely to get such an average.

As for a reason or proof as to why, that is beyond me.

Partial answer - no sum of digits of a power of 2 is divisible by 3. All are divisible by primes higher than three, and (if even) two itself).

### paxwill says

A number's digits will add up to a multiple of 3 if and only if the number has 3 as a prime factor.

http://en.wikipedia.org/wiki/Divisibility_rule

Since 2 is the only prime factor of a power of 2, the sum of digits in a power of 2 cannot equal a multiple of 3, such as 30. If you want to see the list of sums up to 2^10000:

http://oeis.org/A001370/b001370.txt

The digit sums of 2^n are interesting because if you compute their remainders when divided by 9, you get the repeating pattern 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1... As Sid noticed, the digit sum of 2^n is not a strictly increasing function of n. However, the growth is semi-linear and you can estimate the sum of digits in 2^n with the approximation formula 1.355565n. For a graph, see

Thanks, Paxwill. I didn't know the right terms such as "divisibility rule" to research this further. I guessed modular math would be part of it. Is there a name for the branch of math from which divisibility rules are developed?

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