pq^2
A pair of consecutive numbers whose prime factorization is of the same form is called a Corinne Pair.
{21,22} is an example of a Corinne Pair of the form pq because 21 = 3x7 and 22 = 2x11.
{33,34,35} is a Corinne Triple of this form as the prime factors of these three numbers are 3x11, 2x17 and 5x7.
{44,45} is the smallest Corinne Pair of the form pq^2 ... 44 = 11x2^2 and 45 = 5x3^2.
What is the smallest number in the smallest Corinne Triple of the form pq^2.
The answer is 603.
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1 solution
Each of the three consecutive numbers have a factor which is the square of a prime number in it.
So we are looking for a set of consecutive numbers which divide by a selection of 4, 9, 25, 49, 121, 169, 289, 361, etc.
When these divisions are done the result will be three (odd) prime numbers.
Three consecutive numbers will always contain at least one which is even, so we know that one of the Corinne Triple will be a multiple of 4 (= 2^2).
As the other two will both be odd (an odd square multiplied by an odd prime), the multiple of four will be in the middle of the Corinne Triple.
So we are actually searching for a pair of odd numbers of the form 4n-1 and 4n+1, both divisible by odd squares.
Producing a comprehensive list of all multiples of 9, 25, 49 and so on would probably not be too difficult and they could be searched by computer too.
But taking a chance and seeing how the multiples of 9 (by an odd prime) turn out seems a do-able way to start manually.
The following fail for the lack of appropriate neighbours for a variety of reasons - 63, 99, 117, 153, 171, 207, 261,279, 333, 369, 387, 423, 477, 531 and 549.
But 67x9 = 603 has 604 = 151x2^2 for its immediate upper neighbor and continuing in that direction 605 = 5x11^2.
So {603,604,605} is a Corinne Triple of the form pq^2.
Using this as the upper bound for our search with multiples of 25, 49 etc, we quite quickly determine that it is in fact the only Corinne Triple of this form less than 605.