There are 332 odd composites < 1000.

Check all combinations of this set taken 3 at a time; the sums will all be odd.  Will use "combinations_with_replacement" to account for duplicates.

There are 481 positive odd integers between 0 and 999 inclusive which can be written as sum of 3 odd composite positive integers.

This includes all but 5 of the 7 smallest odd composites; in addition, it includes 154 odd primes.

---------------

from itertools import combinations_with_replacement

oddcomposites = [n for n in range(3,1000,2) if not isprime(n)]

ans = []

for comb in combinations_with_replacement(oddcomposites, 3):

    sum3 = sum(comb)

    if sum3 > 999:

        continue

    if sum3 not in ans:

        ans.append(sum3)

print(len(ans))