f(3) = 171. The conjecture is false.

edit: This answer was for the original version, which said the "New Mersenne Conjecture" and gave the formula $f(k)=\frac{2^{2^k+1}+1}{3}$.

f(3) = 171. The conjecture is false.

edit: This answer was for the original version, which said the "New Mersenne Conjecture" and gave the formula $f(k)=\frac{2^{2^k+1}+1}{3}$. At least that version gives integer results; the new one isn't an integer for any k.

f(3) = 171. The conjecture is false.

f(3) = 171. The conjecture is false.

edit: This answer was for the original version, which said the "New Mersenne Conjecture" and gave the formula $f(k)=\frac{2^{2^k+1}+1}{3}$.