A Triangular Array of the Counts of Natural Numbers with the Same Number of Prime Factors (Dimensions) Within 2n Space

Abstract: By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^n+1 (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a dimensions distribution directly related to the distribution of prime numbers. A triangle similar to Pascals triangle from some aspects can be obtained by the ensemble of dimensions distributions. Some superficial explorations have been done to this interesting triangle.