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Linear Combinations of Semiprimes

Problem 278

Published on Saturday, 13th February 2010, 05:00 am; Solved by 613

Given the values of integers 1 < a1 < a2 <... < an, consider the linear combination
q1a1 + q2a2 + ... + qnan = b, using only integer values qk ≥ 0.

Note that for a given set of ak, it may be that not all values of b are possible.
For instance, if a1 = 5 and a2 = 7, there are no q1 ≥ 0 and q2 ≥ 0 such that b could be
1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18 or 23.
In fact, 23 is the largest impossible value of b for a1 = 5 and a2 = 7.
We therefore call f(5, 7) = 23.
Similarly, it can be shown that f(6, 10, 15)=29 and f(14, 22, 77) = 195.

Find ∑ f(p*q,p*r,q*r), where p, q and r are prime numbers and p < q < r < 5000.