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) …