[seqfan] A sequence and a related array

[Ali Sada]

Hi everyone,

If we want to make sure that we have a multiple of a certain positive integer n we can simply multiply n consecutive integers. For example, multiplying 11 consecutive numbers will certainly give us a multiple of 11. 
However, some numbers don’t need n terms to get that multiple. For example, we can get a multiple of 6 by multiplying only three integers m(m+1)(m+2), which means that a(6) = 3. Or if we want a multiple of 7 we need only five terms m(m+1) (m-1)(m^2+m+1)(m^2-m+1). So, what is the least number of these non-factor-able terms we need to multiply in order to get a multiple of n? I would really appreciate your help with this sequence if you thought it’s suitable for the OEIS.

The related array is “the largest common factor of m^k-m^n, where m > 1, n = 1,2,3,.., and k > n."
For example, the largest common factor of m^8-m^2 is 252. I would appreciate your help with this one too.

Best,

Ali