Ryan's Poly-Sum-Deficiency?

Computer Science Level 4

The aliquot sum \(s(n)\) of a natural number \(n\) is the sum of its proper divisors; in other words, the sum of its divisors subtracted by \(n\). For example, \(s(6) = 1+2+3 = 6\).

A natural number \(n\) is deficient if \(s(n) < n\). For example, \(6\) is not deficient since \(s(6) = 6 \ge 6\), but \(8\) is deficient since \(s(8) = 7 < 8\).

A natural number \(n\) is triangular if there exists a natural number \(m\) such that \(n = \frac{m(m+1)}{2}\). Note that \(0\) is not considered triangular.

A natural number \(n\) is a 3-Ryan number if both \(n\) and \(s(n)\) are distinct triangular numbers. For example, \(3\) is a 3-Ryan number since \(3\) and \(s(3) = 1\) are distinct triangular numbers, but \(6\) isn't since \(6\) and \(s(6) = 6\) are not distinct, \(10\) isn't since \(s(10) = 8\) is not a triangular number, and \(4\) isn't since \(4\) is not a triangular number.

Determine the 7th smallest deficient 3-Ryan number.


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