No. Every third consecutive natural number is divisible by 3.
111, 112 and 113.
There must be three consecutive integers to guarantee that the product will be divisible by 6. For the "Product of three consecutive integers..." see the Related Question below.
9240 is the product of the three consecutive integers 20, 21, and 22.
Any two consecutive numbers must comprise one odd and one even number, so their product must be even. Any three consecutive numbers must include two consecutive numbers so the result still applies.
If you define a rectangular number as a number which is the product of two consecutive integers, none of them qualify.
The three consecutive numbers whose sum and product are the same are 1, 2, and 3.