Let $\mathcal S$ be the arithmetic function:

$\displaystyle \mathcal S \left({N}\right) = \prod_{p \mathop \nmid N} \left({1 + \frac 1 {\left({p - 1}\right)^3} }\right) \prod_{p \mathop \backslash N} \left({1 - \frac 1 {\left({p - 1}\right)^2} }\right)$

where:

$p$ ranges over the primes
$p \nmid N$ denotes that $p$ is not a divisor of $N$
$p \mathop \backslash N$ denotes that $p$ is a divisor of $N$.

Let:

$\displaystyle r \left({N}\right) = \sum_{p_1 + p_2 + p_3 \mathop = N} 1$

where $p_1, p_2, p_3$ are prime.

Then for sufficiently large odd integers $N$:

$r \left({N}\right) = \mathcal S \left({N}\right) \dfrac {N^2} {2 \left({\log N}\right)^3} \left({1 + \mathcal O \left({\dfrac{\log \log N} {\log N}}\right)}\right)$

where $\mathcal O$ denotes big-O notation.