#include <utility>
#include <vector>

std::vector< std::pair<int, int> > factor_table;
void fill_sieve( int n )
{
	factor_table.resize(n+1);
	for( int i = 1; i <= n; ++i )
		factor_table[i] = std::pair<int, int>(i, 1);
	for( int j = 2, j2 = 4; j2 <= n; (j2 += j), (j2 += ++j) ) {
		if (factor_table[j].second == 1) {
			int i = j;
			int ij = j2;
			while (ij <= n) {
				factor_table[ij] = std::pair<int, int>(j, i);
				++i;
				ij += j;
			}
		}
	}
}

std::vector<unsigned> powers;

template<int dir>
void factor( int num )
{
	while (num != 1) {
		powers[factor_table[num].first] += dir;
		num = factor_table[num].second;
	}
}

template<unsigned N>
void calc_combinations(unsigned (&bin_sizes)[N])
{
	using std::swap;

	powers.resize(0);
	if (N < 2) return;

	unsigned& largest = bin_sizes[0];
	size_t sum = largest;
	for( int bin = 1; bin < N; ++bin ) {
		unsigned& this_bin = bin_sizes[bin];
		sum += this_bin;
		if (this_bin > largest) swap(this_bin, largest);
	}
	fill_sieve(sum);

	powers.resize(sum+1);
	for( unsigned i = largest + 1; i <= sum; ++i ) factor<+1>(i);
	for( unsigned bin = 1; bin < N; ++bin )
		for( unsigned j = 2; j <= bin_sizes[bin]; ++j ) factor<-1>(j);
}

#include <iostream>
#include <cmath>
int main(void)
{
	unsigned bin_sizes[] = { 5, 15 };
	calc_combinations(bin_sizes);
	char* sep = "";
	for( unsigned i = 0; i < powers.size(); ++i ) {
		if (powers[i]) {
			std::cout << sep << i;
			sep = " * ";
			if (powers[i] > 1)
				std::cout << "**" << powers[i];
		}
	}
	std::cout << "\n\n";
}