Further Observations of Terminal Value Distribution in the Process of Successively Summing Up All Prime Factors of a Given Natural Number

Hung-ping Tsao (2026). Further Observations of Terminal Value Distribution in the Process of Successively Summing up all Prime Factors of a Given Natural Numbers. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Lawrence K. Wang and Hung-ping Tsao (editors). Volume 8, Number 6E, June 2026; 7 pages. Lenox Institute Press MA, USA. ....... ABSTRACT: It is known that each natural number greater than 1 can be expressed as a unique product of prime numbers, namely the prime factorization. For example, 9=3×3 and 3+3=6, we can continue the same process to finally come up with 6=2×3 and 2+3=5, which is a prime number. Accordingly, we obtain 5 to be the terminal value of 9, denoted as t(9)=5. In relating to the above process of successively summing all prime factors of a given natural number, I have recently made the following observation in addition to those in 1. Observations: In the following 5 specifically chosen categories of sequences, the percentage of 5, 7, 11 and 13 each appears as terminal value is quite comparable to that of the natural sequence N(n) up to 1000 terms: 1. Binomial coefficients: C(n,k); 2. Sums of kth powers of natural numbers: S(n,k); 3. Sections divided in order of π-sequence into multiples of m: P(n,m); 4. Sections divided in order of e-sequence into multiples of m: E(n,m); and 5. Sections divided in order of square root-sequence into multiples of m: SR(n,m).