Unique Partition

$5=1+4=2+3$ Let $$w(x)$$ denote the number of ways to express a positive integer $$x$$ as the sum of 1 or more positive integers, all of which are distinct.

For example, $$w(5) = 3,$$ as shown above. Note that the order doesn't matter, so $$(1+4) =(4+ 1)$$ are considered the same thing.

Find the sum of all value of $$a>1$$ such that $$w(a) = a$$.

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