3

This number is a prime.

2^{2^1}-1=3; 2^{2^2}-1 can be written as the sum of a prime power and two distinct powers of two in exactly 3 ways (namely, 2+2^2+3^2, 2^2+2^3+3, 2+2^3+5); 2^{2^3}-1=2^2+2^3+3^5=2+2^7+5^3. For any integer n>3, 2^{2^n}-1 cannot be expressed as the sum of a prime power and two distinct powers of two. [Zhi-Wei Sun]