A number theory problem by Prem Chebrolu

Let $n$ be an integer that cannot be written as the sum of two odd composite numbers. If $n > 33$ , then $n-9, n-15, n-21, n-25, n-27$ and $n-33$ must all be prime [or $n-33 = 1$ , which yields $n = 34 = 9 + 25$ , which doesn't work]. Thus $n-9, n-15, n-21,n-27$ and $n-33$ by forming prime quintuplet. However only one prime quintuplet exists as exactly one of those 5 numbers must be divisible by 5. This prime quintuplet is 5, 11, 17, 23 and 29, yielding a maximum answer of 38. Since $38 - 25 = 13$ , which is prime, the answer is $\boxed{38}$