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
Creators: name: Parker, Alex
name: Tsao, Hung-ping
name: Wang, Lawrence K
Publisher: name: Lenox Institute Press, MA, USA. Lenox.Institute@gmail.com
Year: 2024
Subjects: subject: Factor tables
subject: Mathematics
subject: FOS: Mathematics
subject: Numbers, Prime

description: We present here two more marvelous distribution models M4 and M5 (in addition to M1, M2 and M3 presented in [1]) of the small prime numbers 5, 7, 11, 13 as terminal values in the new platform introduced in [6], 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, …}. Our study of the distribution model M4 for n = 100000 shows that the frequency for each small prime number to occur as terminal values remains almost the same as n increases from 1 to 100000, namely 28% for 5, 16% for 7, 8% for 11 and 9% for 13. We find an estimator E4(n) for the frequencies of the small primes as follows. Let D(n,5) denote the frequency of 5 in a row of terminal values from either A or B for n, with m in a row counted as m-4 number of 5 in a row for m > 5 (for example, one 7 in a row is counted as three 5 in a row's) and let C(n,k) denote the frequency of k in a row of terminal values from C for n. By defining E4(n)=[1E4(n)+2E4(n)]/2, where 1E4(n)=D(n,5)/[D(n,5)+C(n,7)] and 
2E4(n)=D(n,5)/[D(n,5)+C(n,8)], we can come up with excellent estimation. We also come up with a good way of estimating the frequency of singles from A and B combined by using twin primes (tp, tp+2) which will be called cheese sandwich if the terminal value of tp+1 is from C or ham sandwich otherwise, as summarized below for M5 (even n, n = 100000). Moreover, defining E5(n)=[1E5(n)+2E5(n)]/2, where 1E5(n)=D(n,5)/[D(n,5)+C(n,6)] and 2E5(n)=D(n,6)/[D(n,6)+C(n,7)], we can come up with additional good estimation
description: We present here two more marvelous distribution models M4 and M5 (in addition to M1, M2 and M3 presented in [1]) of the small prime numbers 5, 7, 11, 13 as terminal values in the new platform introduced in [6], 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, …}. Our study of the distribution model M4 for n = 100000 shows that the frequency for each small prime number to occur as terminal values remains almost the same as n increases from 1 to 100000, namely 28% for 5, 16% for 7, 8% for 11 and 9% for 13. We find an estimator E4(n) for the frequencies of the small primes as follows. Let D(n,5) denote the frequency of 5 in a row of terminal values from either A or B for n, with m in a row counted as m-4 number of 5 in a row for m > 5 (for example, one 7 in a row is counted as three 5 in a row's) and let C(n,k) denote the frequency of k in a row of terminal values from C for n. By defining E4(n)=[1E4(n)+2E4(n)]/2, where 1E4(n)=D(n,5)/[D(n,5)+C(n,7)] and 
2E4(n)=D(n,5)/[D(n,5)+C(n,8)], we can come up with excellent estimation. We also come up with a good way of estimating the frequency of singles from A and B combined by using twin primes (tp, tp+2) which will be called cheese sandwich if the terminal value of tp+1 is from C or ham sandwich otherwise, as summarized below for M5 (even n, n = 100000). Moreover, defining E5(n)=[1E5(n)+2E5(n)]/2, where 1E5(n)=D(n,5)/[D(n,5)+C(n,6)] and 2E5(n)=D(n,6)/[D(n,6)+C(n,7)], we can come up with additional good estimation