I have a list of numbers and I want to express each one as an integer linear combination of $2^{(2^k)}$ powers.
Some elements of the list are
(3748, 64594, 16700300, 68685479824, 4722365919717929558384, ...)
The first two elements can be expressed for example as
$$ 3748 = 2^2 - 6\cdot2^4 + 15\cdot2^8 $$
and
$$ 16700300= - 2^2 + 9\cdot2^4 - 45\cdot2^8 + 255\cdot2^{16}, $$
where the exponents $2$, $4$, $8$ and $16$ are all powers of 2.
Can you help me write a program which yields such coefficients? the answers must be of the form ...+-+-+-...
3748 ===> {1, -6, 15}
16700300 ===> {-1, 9, -45, 255}
I also realised that if the prime factors of the number are not raised to any power then you can find the coefficients here (i.e. 2*3*5*7 has n=4)
n= number of distinct primes
Table[(-1)^(n - k) Binomial[n, k], {k, 0, n}]
Is there a formula for factors raised to powers?