15. Prime tuples

Mathematicians like playing with prime numbers, so they have come up with several different names for groupings of related primes:

• Twin primes are primes {p, p + 2} that differ by two, such as {3, 5}. (Primes that differ by only 1 are not very interesting because 2 and 3 are the only primes that differ by 1.)
• Cousin primes are primes {p, p + 4} that differ by 4, such as {3, 7}.
• Sexy primes are primes {p, p + 6} that differ by 6, such as {5, 11}. (The name sexy primes is a pun because sex is Latin for six.)

You can also look for different numbers of primes with various spacings. For example, you can look for sexy pairs, sexy triples such as {7, 13, 19}, and so forth.

Write a program that checks numbers up to a maximum value, looking for primes with a given spacing and quantity. For example, the user might set the spacing to six and the number to 3 to look for groups of three primes that are each six apart, such as {5, 11, 17}.

Use your program to see what's special about groups of three or four primes that differ by 6, 12, 18, and other multiples of 3.