Distribution of squares and hypothesis verification in odd integer partitions

The article considers partitions of a natural number $n$ whose parts are distinct, odd and their product is not a square. Such partitions are applicable for determining the rank of the group of central units of an integral group ring of an alternating group. The number of partitions grows exponentially, hence the enumeration task is computationally expensive. The article proposes a parallel algorithm in shared memory for finding the number of partitions of the $n$ with additional conditions. The algorithm is based on the concept of data parallelism and the use of nested parallelism. A set of lengths $K$ of the partition of the $n$ is obtained, the elements of which are processed in parallel. During the processing of the length $k$ of the partition of the $n$, the set of levels $L$ is obtained, the elements of which are processed in parallel as well. Acceptable values of speedup and parallel efficiency of the proposed algorithm are obtained by using two threads per parallel length region and two threads per parallel level region. Thus, the speedup for distinct $n$ increases to $2.1$, and the parallel efficiency does not decrease below $50\%$. The results obtained were used to check Kargapolov's hypotheses and analyze the distribution of the number of odd partition in certain ranges. An algorithm for finding the optimal coefficient $c$ is proposed. Using this algorithm, an asymptotic formula for the number of partitions of the $n$ is obtained, in which the parts are distinct, odd and their product is a square. This formula is based on experimental data and formulated as a conjecture.