For each positive integer $n$, let $q(n)$ denote the number of ways to write $n$ as a sum of distinct positive integers. We call those $q(n)\ (n=1,2,3,\ldots)$ strict partition numbers. The sequence $q(1),q(2),q(3),\ldots$ is available from http://oeis.org/A000009. It is known that $$q(n)\sim\frac{e^{\pi\sqrt{n/3}}}{4(3n^3)^{1/4}}\qquad \ \text{as}\ \ n\to+\infty.$$
In 1996 P. Erdos and M. Lewin [Math. Comp. 65 (1996), 837-840] proved that each positive integer can be written as a sum of some elements of the set $\{2^a3^b:\ a,b=0,1,2,\ldots\}$ with no one dividing another. Motivated by this, we formulate the following conjecture.
Conjecture. Each positive integer $n$ can be written as a sum of strict partition numbers $q_1,\ldots,q_k$ such that $q_i\nmid q_j$ for all $i,j=1,\ldots,k$ with $i\not=j$.
I have verified this for all $n=1,\ldots,1350$. For example, \begin{align*}1344&=8+76+1260,\\ 1345&=10+222+1113, \\1346&=12+18+38+296+982\end{align*} with $8,10,12,18,38,76,222,296,982,1113,1260$ strict partition numbers.
QUESTION. Can one find a counterexample to the above conjecture? Any possible way to prove the conjecture?
Your comments are welcome!
