A positive integer N is such that:

(i) N is divisible by all but three of the
k positive integers 1, 2, 3, . . . , k; and:

(ii) Those three positive integers (that don’t divide N) are consecutive.

(For example, if N=60 then the corresponding value of k is 10, since 7, 8, 9 doesn't divide 60.)

Determine (with proof) the maximum value of the positive integer k satisfying all the given conditions.