writing an integer as particular summation [closed]

Question

I think my question is an elementary question. Thanks for any help or comment.

Is there any formula for the number of writting a natural number $n$ in a summation as follows,

$n=a_1+\dots+a_k$, where $a_i>1$ and $a_i\neq a_j$

for example suppose $n=10$, then there is 5 types for writing $10$ as above $10=2+8=3+7=4+6=2+3+5=10$

Answer 1

Generating function is $\prod_{k\geqslant 2} (1+x^k)$. It allows to get asymptotics of Hardy-Ramanujan-Rademacher type. Also we may express it as $P(n)-P(n-1)+P(n-2)-\dots$, where $P$ denotes number of partitions into different parts.