Below, is the definition of a **prime-partitionable** integer taken from W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272 and is apparently the same in W. T. Trotter, Jr. and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142 DOI:10.1002/jgt.3190020206:

An integer $n>=2$ is said to be prime-partitionable if there is a partition {$P_1,P_2$} of the set $P$ of all primes less than $n$ such that, for all natural numbers $n_1$ and $n_2$ satisfying $n_1+n_2=n$ we have that either $gcd(n_1,p_1) \ne 1$ or $gcd(n_2,p_2) \ne 1$ or both, for some pair $(p_1,p_2) \in P_1 \times P_2$.

Conjectures: If $P_1 =$ {$p_{1a}, p_{1b}$}, $p_{1a}$ and $p_{1b}$ are odd primes and $p_{1a}< p_{1b}$, it appears that, if $\psi = kp_{1a} + 1 = p_{1b} + p_{1a}$ for $k$ odd and $1 < k ≤ p_{1a}-2 $ then:

$\psi$ is prime-partitionable,

no two values of $p_{1b}$ are the same and

the number of values of $k$ is $≥ 1$ for each $p_{1a} ≥ 5$.

The following six sequences in the OEIS contain the results of my investigations into prime-partitionable numbers and their equivalence to Erdös-Woods numbers:

A059756, A244640, A245664, A249302, A245372, A259560

Can proofs be found for the conjectures please.