[seqfan] Can we always reach a power of 2 with this formula?

[Ali Sada]

Hi everyone,

a(n) is the least positive integer such that 2^n + a(n)*A002378(n) equals to a power of 2. 

2^1 + 1*2 = 4,  a(1) = 1
2^2 + 2*6 = 16,   a(2) = 2
2^3 + 2*12 = 32,   a(3) = 2
2^4 + 8*20 = 256,  a(4) = 8
2^5 + 16*30 = 512, a(5) = 16
2^6 + 96*42 = 4096, a(6) = 96
2^7 + 16*56 = 1024, a(7) = 16 

Can we prove that we can always find a value for a(n)?
I would like to add this sequence to the OEIS and I would really appreciate it if someone could calculate the necessary terms if possible. 

Best,

Ali