Prime Problem (reposted)

Number Theory Level 5

Let \(n\) denote the product of the first 2013 primes. Find the sum of all primes \(p\) (for \(20<p<150\)) such that

  • \(\dfrac{p+1}{2}\) is even but is not a power of 2.

  • there exist pairwise distinct positive integers \(a,b,c\) for which \[a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b)\] is divisible by \(p\), but not by \(p^2\).

This is part of Ordered Disorder.

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