Odd numbers expressible as a sum of 3 composite odd integers. How many positive odd integers between $0$ and $999$ inclusive can be
written as sum of $3$ odd composite positive integers?

I am trying to approach this by looking at equivalence classes $\mod 6$ but the calculations are a bit tedious.
Let's start with integers equivalent to $3\pmod6$.  The smallest is obviously $27=9+9+9$.  Every following odd multiple of $3$ also works as it can be written as $9+9+3(2n+3)$.  $27=6(4)+3$ and $999=6(166)+3$, so we have $163$ odd multiples of $3$.

The first odd number not a multiple of $3$ is $5(5)=25\equiv1\pmod6$.  So the first value equivalent to $1\pmod6$ is $9+9+25=43=6(7)+1$.  The same trick in the first step works here, so we have $160$ values here.

The first odd number equivalent to $5\pmod6$ is $5(7)=35$.  The smallest value that works equivalent to $5\pmod6$ is then $9+9+35=53&lt;9+25+25$.  $53=6(8)+5$.  We have $158$ numbers here that work.

So our final tally is $163+160+158=481.$
Let the three odd numbers be $2a + 1$, $2b + 1$, $2c + 1$.

The sum of these numbers is $6(a + b + c) + 3$. Our goal is to find out the number of integers of that form. 

Equating it with 999, we find $a + b + c$ equals 166. This implies there should exist 166 odd numbers that are the sum of three odd numbers.