On the parity of the number of partitions in square free parts (Q1417941)

Let \(\tilde{p}(n)\) denote the number of partitions of a positive number \(n\) into square-free parts. The author studies the parity of \(\tilde{p}(n)\) using differential equations in the ring of formal power series \(F_2[[X]]\), where \(F_2\) is the field with two elements. He proves that for large \(N\) we have \[ \# \{n \leq N: \tilde{p}(n) \equiv 1 \pmod{2} \} \geq c_1 \log N \] and \[ \# \{n \leq N: \tilde{p}(n) \equiv 0 \pmod{2} \} \geq \frac{c_2N}{\log N}, \] where \(c_1\) and \(c_2\) may be taken to be any positive numbers less than \( \frac{1}{2\log 2}\) and \(\frac{\log 2}{6}\), respectively.