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Common factors between two sequences

Published on Sunday, 16th November 2014, 04:00 am; Solved by 161;
Difficulty rating: 100%

Problem 489

Let G(a, b) be the smallest non-negative integer n for which gcd(n3 + b, (n + a)3 + b) is maximized.
For example, G(1, 1) = 5 because gcd(n3 + 1, (n + 1)3 + 1) reaches its maximum value of 7 for n = 5, and is smaller for 0 ≤ n < 5.
Let H(m, n) = Σ G(a, b) for 1 ≤ am, 1 ≤ bn.
You are given H(5, 5) = 128878 and H(10, 10) = 32936544.

Find H(18, 1900).