2010/036) if n is an integer give an example of n consecutive composite integers

in the 1st attempt the solution appears to be

(n+1)! + 2 to (n+1)!+n+1 the kth term is divisible k+1

but it can be bettered as

LCM(2,3,... n+1) + 2 to
LCM(2,3,... n+1) + n+ 1

and LCM(2,3,... n+1) is the lower of the two

It can still be bettered to the product of all primes (<= n+1)say P(n) + 2 to that + n+ 1

as the number from 2 to n+1 if a prime the product has that prime as a factor

and if not a prime then P(n) and and k have a common factor so sum is not a prime

(one of my friends informed me)