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Problem 278

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Linear Combinations of Semiprimes

Given the values of integers 1 < a₁ < a₂ <… < a_n, consider the linear combination q₁a₁ + q₂a₂ + … + q_na_n = b, using only integer values q_k ≥ 0.

Note that for a given set of a_k, 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.

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半素数线性组合

给定整数1 < a₁ < a₂ <… < a_n，考虑其线性组合q₁a₁ + q₂a₂ + … + q_na_n = b，其中q_k ≥ 0均为整数。

注意到，对于特定集合a_k，不是所有的b值都能取到。
例如，如果a₁ = 5且a₂ = 7，不存在q₁ ≥ 0和q₂ ≥ 0使得b取1、2、3、4、6、8、9、11、13、16、18或23。
事实上，当a₁ = 5且a₂ = 7时，不能取得的b值最大为23。
我们因此记f(5, 7) = 23。
类似地可以得出f(6, 10, 15)=29以及f(14, 22, 77) = 195。

对于所有素数p < q < r < 5000，求∑ f(p*q,p*r,q*r)。

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