Let's define L(n, k) as the largest prime factor of product

n*...*(n+k)

of k+1 consecutive integers, starting at positive integer n.

So we have, for example,

L(4374, 1) = 7

L(48, 2) = 7

L(350, 2) = 13

L(138982582998, 2) = 103

L(61011223, 3) = 163

L(23931257472314, 3) = 631

L(1517, 4) = 41

L(3294850, 5) = 239

L(1913253200, 8) = 3499

L(8559986129664, 12) = 58393

L(48503, 14) = 379

**Conjecture:**
as n goes to infinity,

lim inf L(n, k) / (log n)^2 = C_k

The very rough estimates of constants C_k are:

C_1 ~ 0.0376

C_2 ~ 0.258

C_3 ~ 0.907

C_4 ~ 2.46

C_5 ~ 5.16

C_6 ~ 11.4

C_7 ~ 19

C_8 ~ 42

C_9 ~ 70

C_10 ~ 140

C_11 ~ 200

C_12 ~ 250

C_13 ~ 380

C_14 ~ 430

C_15 ~ 460

Some successive minima of L(n, k) are shown there:

http://oeis.org/A193943
http://oeis.org/A193944
http://oeis.org/A193945
http://oeis.org/A193946
http://oeis.org/A193947
http://oeis.org/A193948
http://oeis.org/A199407
http://oeis.org/A200566
http://oeis.org/A200567
http://oeis.org/A200568
http://oeis.org/A200569
http://oeis.org/A200570
http://oeis.org/A209837
http://oeis.org/A209838
http://oeis.org/A209839
Any suggestions on the conjecture? Does it depend on other

known ones like Twin prime conjecture or ABC conjecture?

Great thanks for any information on the subject.