The Powers of \(32\)!

Number Theory Level 3

One day for no apparent reason, Brilli the ant decides to add up different non-negative powers of \(32\) and begins to write them down in ascending order. The first few terms of her sequence look like this:

\[1, 32, 33, 1024, 1025, 1056, 1057, 32768, 32769, 32800, 32801...\]

If \(35467067393\) is the \(N^{\text{th}}\) term of this sequence, what is \(N\)?

Details and assumptions:

A term can be the sum of more than two different powers of \(32\). For example: \(32801=32^3+32^1+32^0\).

A term can be the "sum" of one number. For example: \(1024=32^2\) is a term of this sequence.

A term, however, can't be the "sum" of zero numbers. In other words, \(0\) isn't in this sequence.

A term also has to be the sum of distinct numbers. That means \(65=32^1+32^1+32^0\) is not in the sequence.

This problem was inspired by a problem posed in the BdMO-2013 divisional round.


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