Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often

The infinite sequence of gaps (first differences) between successive odd composite numbers contains only the numbers 2, 4, and 6. We prove that, for any natural number k, the sequence of gaps contains infinitely many k-tuplets of consecutive gaps all equal to 2. Infinitely many gaps equal 4. The sequence of gaps includes infinitely many gap pairs (4,4) if the sequence of positive primes has infinitely many pairs of successive primes that differ by 4 (cousin primes), which is unproved but holds under a conjecture of Hardy and Littlewood. Gap triplets (4,4,4) never occur. Infinitely many gaps equal 6 if and only if there are infinitely many twin primes. Moreover, gap pairs (6,6) occur infinitely often if other conjectures of Hardy and Littlewood are true. Six of the 27 potential triplets of values of gaps between successive odd composite numbers never occur: (4,4,4), (6,6,6), (6,4,4), (4,4,6), (6,2,6), and (6,4,6).