ternary mekso operator: x_{1}th Bergelson multiplicative interval with exponents bounded from above by function x_{2} and with sequence of shifts x_{3}, where exponents belong to set x_{4}

x1 must be a positive integer. x2 must be a strictly monotonic increasing function mapping from all of the positive integers to a subset (not necessarily proper) thereof. x3 must be a sequence of natural numbers. x2 without context will default to the same value as x1 (it is simple linear on the set of positive integers), x3 without context will be a sequence all and only of 1's, x4 without context defaults to the set of all non-negative integers. Let p_i be a prime for all i, with p_1 = 2 and the ith prime (in the normal monotonic increasing order) being p_i. Let all other symbols match the aforementioned conditions. Represent the nth term of the sequence x3 by x3_n; represent the function in x2 being applied to the number m by x2(m). Then x1 be'ei'oi x2 boi x3 boi x4 produces the set of all numbers of the form x3_(x1) * (p_1)^(e_1) *...* (p_(x1))^(e_(x1)), where e_j belongs to the intersection of the interval [0, x2(x1)] with x4.