# Problem 333: Special partitions
All positive integers can be partitioned in such a way that each and
every term of the partition can be expressed as 2ix3j, where i,j ≥ 0.
Let's consider only those such partitions where none of the terms can
divide any of the other terms. For example, the partition of 17 = 2 + 6
+ 9 = (21x30 + 21x31 + 20x32) would not be valid since 2 can divide 6.
Neither would the partition 17 = 16 + 1 = (24x30 + 20x30) since 1 can
divide 16. The only valid partition of 17 would be 8 + 9 = (23x30 +
20x32). Many integers have more than one valid partition, the first
being 11 having the following two partitions. 11 = 2 + 9 = (21x30 +
20x32) 11 = 8 + 3 = (23x30 + 20x31) Let's define P(n) as the number of
valid partitions of n. For example, P(11) = 2. Let's consider only the
prime integers q which would have a single valid partition such as
P(17). The sum of the primes q <100 such that P(q)=1 equals 233. Find
the sum of the primes q <1000000 such that P(q)=1.