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NT, p_1 + p_2 + ...+ p_n = c^2_{n-1}

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 Shortlist N5 day2, C2 p2

September 30, 2021
number theory

Problem Statement

A positive natural number nn is said to be comico if its prime factorization is n=p1p2...pkn = p_1p_2...p_k, with k≥3k\ge 3, and also the primes p1,...,pkp_1,..., p_k they fulfill that p1+p2=c12p_1 + p_2 = c^2_1 p1+p2+p3=c22p_1 + p_2 + p_3 = c^2_2 ...... p1+p2+...+pn=cn−12p_1 + p_2 + ...+ p_n = c^2_{n-1} where c1,c2,...,cn−1c_1, c_2, ..., c_{n-1} are positive integers where c1c_1 is not divisible by 77. Find all comico numbers less than 10,00010,000.
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