  A positive integer is called an $E_j$-number if it is the product of $j$
distinct primes. We prove that there are infinitely many triples of
$E_2$-numbers within a gap size of $32$ and infinitely many triples of
$E_3$-numbers within a gap size of $15$. Assuming the Elliot-Halberstam
conjecture for primes and $E_2$-numbers, we can improve these gaps to $12$ and
$5$, respectively. We can obtain even smaller gaps for almost primes, almost
prime powers, or integers having the same exponent pattern in the their prime
factorizations. In particular, if $d(x)$ denotes the number of divisors of $x$,
we prove that there are integers $a,b$ with $1\leq a < b \leq 9$ such that
$d(x)=d(x+a)=d(x+b) = 192$ for infinitely many $x$. Assuming Elliot-Halberstam,
we prove that there are integers $a,b$ with $1\leq a < b \leq 4$ such that
$d(x)=d(x+a)=d(x+b) = 24$ for infinitely many $x$.
