Every Odd Number Greater Than 1 is the Sum of at Most 241 Primes

Read-back: odd_sum_le_241_primes

For every natural number $n$ such that

• $n$ is odd, meaning $n = 2k+1$ for some natural number $k$, and
• $n > 1$ (strictly),

there is a finite multiset $s$ of natural numbers with these three properties:

In other words, every odd natural number $n$ greater than $1$ can be written as the sum of at most $241$ primes. Here "prime" is the usual notion for natural numbers: $p \ge 2$ and the only divisors of $p$ are $1$ and $p$.

Fine print:

• Since $s$ is a multiset, the same prime may appear more than once, and $|s|$ counts every copy.
• The order of the summands does not matter.
• The bound is the non-strict $\le 241$, so any number of summands from $1$ to $241$ is allowed. For example, when $n$ is itself prime, the one-element multiset $\{n\}$ works.
• The empty multiset (zero summands, sum $0$) can never be a witness, because $n \ge 3$.
• Together the two hypotheses say exactly that $n$ ranges over $3, 5, 7, 9, \dots$. They can always be satisfied, so the statement is not vacuous.
• The statement only claims that such a multiset exists. It says nothing about the summands being distinct, about how many representations there are, or about any bound smaller than $241$.
• No other variables, assumptions or definitions appear. The statement uses only the standard library notions of oddness, primality, multiset size and multiset sum.