To get some insight into what’s going on here, it may help to know exactly which intervals contain a given number of primes. We can add a few lines to our code in order to output a list of \(a\) for which \((a, a + H]\) contains exactly \(m\) primes, for \(m\) in a given list of interest.
The function is overlap_extension in src/primes_in_intervals/intervals.py (pii overlap-extension from the shell), and it is overlap with a notebook attached. It takes the same \(A\), \(B\), \(H\), together with a list M of values of \(m\) to watch, and runs the same two-generator sliding window. The extra bookkeeping rides on the window’s events: whenever the current count \(m\) is one of the watched values, the left endpoint \(a\) is recorded, and, since the count stays at \(m\) until an endpoint next passes a prime, the whole run of left endpoints from \(a\) up to the next event is recorded with it. The function returns a pair: the dictionary of watched values, {m : [every a with exactly m primes in (a, a + H]]}, and the usual counting dictionary {m : h(m)}, so the lists can be checked against the counts at a glance.
from primes_in_intervals import overlap_extension
overlap_extension(1000,2000,50,[3,11])({3: [1307, 1308, 1309, 1310, 1321, 1322, 1327, 1328, 1329, 1330],
11: [1271, 1272, 1273, 1274, 1275, 1276, 1277, 1278]},
{3: 10, 4: 94, 5: 138, 6: 202, 7: 216, 8: 178, 9: 136, 10: 18, 11: 8})Ten intervals \((a, a + 50]\) with \(a\) between \(1000\) and \(2000\) contain exactly three primes, and the list shows where they all are: four consecutive left endpoints from \(1307\), two more runs around \(1321\) and \(1327\). At the other extreme, the eight intervals containing eleven primes sit in one run, \(a = 1271\) through \(1278\), a neighborhood worth a closer look.
The function is there to help us investigate, and this is as far as we have taken it: we may do more with it later.