4 for fort

Number Theory Level 4

Let \(a_1,a_2,a_3,\ldots,a_n\) be a sequence of all numbers which can be expressed as sums of different non-negative integer powers of 4 (which is the form of \(4^{k_1}+4^{k_2}+4^{k_3}+\ldots+4^{k_p}\), \(k_m\neq k_n\), \(m,n={1,2,3,\ldots,p}\),\(m\neq n\) where \(k\) is a non-negative integer) arranging from the smallest to the largest.

The first few terms are as follows, \[a_1=4^0=1\] \[a_2=4^1=4\] \[a_3=4^0+4^1=5\] \[a_4=4^2=16\] \[a_5=4^0+4^2=17\] \[\ldots\]

Find the value of \(a_{64}\).

(Note: The \( k_i \) need not to be the same for different \( a_j \))

This is one part of 1+1 is not = to 3.


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