  We derive explicit formulas for integrals of certain symmetric polynomials
used in Keiju Sono's multidimensional sieve of $E_2$-numbers, i.e., integers
which are products of two distinct primes. We use these computations to produce
the currently best-known bounds for gaps between multiple $E_2$-numbers. For
example, we show there are infinitely many occurrences of four $E_2$-numbers
within a gap size of 94 unconditionally and within a gap size of 32 assuming
the Elliott-Halberstam conjecture for primes and sifted $E_2$-numbers.
