Amazing Estimators for the Frequencies of Small Prime Numbers as Terminal Values in the Process of Successively Summing Up All Factors of the Natural Numbers Ii

Hung-ping Tsao (2026). Amazing Estimators for the Frequencies of Small Prime Numbers as Terminal Values in the Process of Successively Summing up all Factors of the Natural Numbers II. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM) ", Lawrence K. Wang and Hung-ping Tsao (editors). Volume 8, Number 2, February 2026; 14 pages. Lenox Institute Press MA, USA. . . . . ABSTRACT: We present here yet another new platform for studying the small prime numbers 5, 7, 11, 13 as terminal values, where the terminal value t (n) for each natural number n being defined by way of successively summing up the prime factors. By defining A = 5 and B = 7, 11, 13, as in 1, we call 5 the A-terminal value, 7, 11 or 13 a B-terminal value and all others C-terminal values, where C = 4, 17, 19, 23, 29, …. We shall extend the terminal value of each natural number n to the terminal value sequence ts (n) as follows. t1 (n) =t (n), t2 (n) =t (n+t1 (n) ), t3 (n) =t (n+t1 (n) +t2 (n) ), …, tm (n) =t (n+t1 (n) +t2 (n) +…+tm-1 (n) ), … Procedure to calculate the terminal value sequence of n. (s0=n, s1=s0+t1 (n), s2=s1+t2 (n), …) m 1 2 3 s s0 s1 s2 t1 t (s0) t (s1) t (s2) …