The Laws of the Minor Prime Factors

The Newcomb-Benford Law (NBL) suggests that the smaller digits of significands represented in place-value notation are more likely to appear in real-life numerical datasets. We propose that similar laws exist regarding the prime factorization of these significands. By the fundamental theorem of arithmetic, we can express a natural number as an ordinal-ascending sequence of ordinal-multiplicity pairs representing the prime factors by which N is divisible. We refer to this as the Standard Ordinal-Exponent representation (SOE). The costs of the positional and SOE representations interconnect through the double logarithmic scale; the size of a number written in positional notation has the same order of growth as the exponential of the SOE sequence length. Based on the SOE representation, we submit a battery of laws exhibiting the prevalence of the minor prime powers across the natural numbers, to wit, the probability of a prime relative to the factorization set, the probability and possibility of the smallest prime ordinal, the probability of the number of participants in an interaction (regarding and disregarding multiplicity), the probability and possibility of a prime divisor with multiplicity, the probability of a prime exponent, and the probability of the largest prime exponent. Then, we factorize two NBL-compliant datasets to investigate key properties of primality: a 300-entry dataset comprising mathematical and physical constants (CT), and another containing 1,080 entries of world population data (WP). For both, we examine the energy function E(N)=p_N/N, the omega functions ω(N) (number of distinct prime factors) and Ω(N) (total number of prime factors), the divisor functions d(N) (number of divisors) and σ(N) (sum of divisors), as well as the share of rough-smooth numbers, the growth of highly composite numbers, and the prime-counting π(N) and totient φ(N) functions. Besides, we confirm compliance with the laws above and analyze the internal count of primes, the density of the largest prime ordinal, the internal growth of totatives and non-totatives, the density of k-almost primes, and the distribution of the pairwise greatest common divisor. CT and WP are chunks of nature. Indeed, we can identify natural datasets by testing their conformance to NBL or to any of the criteria we postulate. We also emphasize that the artanh function prominently appears throughout our analysis, suggesting that the concept of conformality governs our perception of the external world, bridging information between the harmonic scale (global) and our logarithmic scale (local).