Irreducible Partitions

[David Harden]

>The asymptotic behavior of the sequence is also far from clear - >except that are
>infinitely 1's.  Do other values occur infinitely often?  I don't even >see any
>way to prove that the sequence is unbounded, although it >certainly seems to be.

Here is a proof that the number of irreducible partitions into squares is unbounded:

Note 1. Suppose n is not a square. Then any partition of n into 2 squares is irreducible.
Note 2. The number of expressions of n as a sum of two squares is easy to specify in terms of the prime factorization of n. In particular, if n is squarefree and n=p_1*...*p_k is the prime factorization of n, then the number of these expressions is 0 if any p_i is congruent to 3 mod 4; otherwise, if n is odd (I am still assuming n to be squarefree), the number is 2^(k-1).
Note 3. There are infinitely many primes congruent to 1 mod 4.

Then let p_n be the nth prime congruent to 1 mod 4. Then p_1*...*p_m has at least 2^(m-1) irreducible partitions into squares since it is not a square, since we can just look at the partitions into two squares. Since we can make m arbitrarily large, the proof of the unboundedness is complete.

Good luck learning more about this sequence.

---- David