What is the next composite number?

If $p_n$ is the nth prime number. then what is the next composite number after say $p_4^2\times p_5$ without actual calculation? ($p_4^2\times p_5+1$ is $p_1^2p_2^3p_3$) the first few composite numbers seem to follow no immediate pattern i.e. $p_1^2,p_1p_2,p_1^3,p_2^2,p_1^2p_2,\cdots$

For example given the consecutive composite numbers $p_{j_1}^{k_1}p_{j_2}^{k_2}\cdots p_{j_n}^{k_n}$ and $p_{m_1}^{n_1}p_{m_2}^{n_2}\cdots p_{m_r}^{n_r}$ what is the mapping from $\big( ({j_1},{k_1}),({j_2},{k_2}) \cdots ({j_n},{k_n})\big) \to \big( ({m_1},{n_1}),({m_2},{n_2}) \cdots ({m_r},{n_r}) \big)$