Number as Sum of Distinct Primes

Theorem
For $n \ne 1, 4, 6$, $n$ can be expressed as the sum of distinct primes.

Proof
Let $S = \set {s_n}_{n \mathop \in N}$ be the set of primes.

Then $S = \set {2, 3, 5, 7, 11, 13, \dots}$.

By Bertrand-Chebyshev Theorem:
 * $s_{n + 1} \le 2 s_n$ for all $n \in \N$.

We observe that every integer $n$ where $6 < n \le 6 + s_6 = 19$ can be expressed as a sum of distinct elements in $\set {s_1, \dots, s_5} = \set {2, 3, 5, 7, 11}$.

Hence the result by Richert's Theorem.

Here is a demonstration of our claim: