
EXAMPLE

a(5) = 3960 is divisible by 8, 9, 10, 11, and 12, but not 7 or 13. It is the smallest positive integer with a string of 5 consecutive divisors that is not part of a longer string.
From Gus Wiseman, Oct 16 2019: (Start)
The sequence of terms together with their divisors begins:
1: {1}
2: {1,2}
6: {1,2,3,6}
12: {1,2,3,4,6,12}
3960: {1,2,...,8,9,10,11,12,...,1980,3960}
60: {1,2,3,4,5,6,...,30,60}
420: {1,2,3,4,5,6,7,...,210,420}
840: {1,2,3,4,5,6,7,8,...,420,840}
(End)


MATHEMATICA

tav=Table[Length/@Split[Divisors[n], #2==#1+1&], {n, 10000}];
Table[Position[tav, i][[1, 1]], {i, Split[Union@@tav, #2==#1+1&][[1]]}] (* Assumes there are no zeros.  Gus Wiseman, Oct 16 2019 *)
