Hung-ping Tsao (2023). A Well-rounded Distribution of Small Prime Numbers as Terminal Values in the Process of Summing up all Factors of the Natural Numbers. In: “Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)”, Lawrence K. Wang and Hung-ping Tsao (editors). 5(10D), October 2023; 59 pp. Lenox Institute Press MA, USA. tsaohp.tsao6@gmail.com ; doi.org/10.17613/aw2z-vs27 …… ABSTRACT: We provide here a new platform for exploring of prime numbers by defining the terminal value t(n) for each natural number n by successively summing up its prime factors. We further define two sets A={5} and B={7, 11, 13}. We shall call 5 the A-terminal value, 7, 11 or 13 a B-terminal value and all others C-terminal values. We have found, up to
n =3002, 1 septuple (7 in a roll) A-terminal values ending at 2330, 1 septuple B-terminal values ending at 1595 and 3 sextuple (6 in a roll) C-terminal values ending at 1551, 2562 and 2986, respectively. We further spot checked around n = 10000 and found 1 nonuple (9 in a roll
C-terminal values ending at 9787. We finally found that 60% (59%, respectively) of terminal values are 5, 7, 11 or 13 up to n = 2000 (3000, respectively) and came up with a good way of estimating such percentage p(n) by using twin primes, which will be called cheese sandwich if the terminal value of the number sandwiched in between twin primes in question is from C or ham sandwich otherwise. As it turned out, the percentage of ham sandwiches among twin sandwiches is a slightly less than p(n) for n = 2000 or 3000. CONJECTURE 1. In a long run, the percentage of single terminal values from A and B combined is slightly less than 60% and the percentages for double, triple and quadruple terminal values from A and B combined are 16%, 4% and 1%, respectively. CONJECTURE 2. There would not be any m-tuple terminal values from A or B for m > 10. CONJECTURE 3. There might be some m-tuple terminal values from C for m > 10.