Not always adorable!

Number Theory Level 3

A natural number \(m\) is adorable if you can write it as sum of three natural numbers \(m = a+b+c\) such that \( (2 \mid a), (3 \mid b), (4 \mid c) \).

Find the sum of all natural numbers which are not adorable.

Details and assumptions:

  • \(a\mid b\) means \(a\) divides \(b\), i.e. \(b\) is divisible by \(a\).

  • As an explicit example, the number \(11\) is adorable because \(11 = 4+3+4\), and \( 2\mid 4 ; 3\mid 3 , 4\mid 4\). But the number \(7\) is not adorable because you can't find such \(a,b,c\).


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