Ramanujan might take a little longer on this one

Computer Science Level 3

This problem is a much more difficult computer-science follow-up to this question. If you have already read the paragraph about Ramanujan there, you can skip it here.

Today, Sunday, December \(22\), is Srinivasa Ramanujan's \(126\text{th}\) birthday. Ramanujan was an amazing mathematician, but one of the things he was most famous for had to do with the number \(1729\). When Ramanujan was in the hospital, he was visited by his friend G.H. Hardy. Hardy remarked that the taxicab that he had ridden in had a rather uninteresting number: \(1729\). Ramanujan said that no, \(1729\) was very interesting because it was the smallest number that can be expressed as the sum of \(2\) cubes in \(2\) different ways. These are \(12^3+1^3\) and \(10^3+9^3\). Hence, numbers that can be written as the sum of multiple cubes in more than \(1\) way are called taxicab numbers. Some more really interesting information about taxicab numbers can be found in this video.

So here's the problem.

Let \(f(n)\) be a function that finds the sum of the digits of \(n\). What are the last \(3\) digits of this expression? \[\sum_{i=1}^{17}\sum_{j=1}^{29} f(i^3+j^3)\]


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