Author – Andrew J Frost 10/06/2020 REV TWO PnonP p2
OBSERVATION 2 Pattern:- Raised by….
The integer series when multiplied by 12 and raised by each of the four prime numbers 1,5,7 and 11, less than 12; produces a pattern corresponding to, 1 5 7 11, 1 5 7 11, 1 5 7 11, 1 5 7 11,….ad infinitum.
Constructing a List
This produces my special list when it also includes numbers divisible by 5.
However, when the spreadsheet is sorted (filtered/sieved). On sieving out of all the resultant numbers in the list of primes and non-primes that end in the digit 5, including the initial solitary 5, the pattern changes.
The ‘Raised By Pattern’
The pattern in this sequence now starts with 1, 7 and 11. The complete pattern of 16 numbers becomes: – 1, 7, 11, 1, 5, 7, 11, 5, 7, 1, 5, 7, 11, 1, 5, 11. These numbers then repeat as a sequence for this whole series of prime/non-prime numbers in the infinite Number-Line (see excerpt from the spreadsheet below).
The series does not vary up to 6011, which is as far as I went initially in calculating the prime factors of each non-prime odd-composite number, at this stage.
This pattern now produces my special Set S(af) as follows.
My logical process formula for the generation of my special Set S(af) comprising prime and non-prime odd-composite numbers: –
12n + {1,5,7,11} = p(p&np)
and p(p&np) –[div5] = S(af)
where: –
- n is an integer
- 1,5,7,and 11 are prime numbers used to raise the resultant of 12n to “p(p&np)” are primes and non-primes evolved from odd-composite numbers.
- [div5] is redundant information (i.e. all odd-composite numbers ending in 5) hence the ‘minus’ which represents a sieve (or sort) to remove them.
S(af) my Set comprises all the primes (subSet ‘primes’) and the non- primes (subSet ‘non-primes’), composed from odd-composite numbers according to the origination process above.
^ So taking the results from my logical process formula above, the prime and non-primes are completely complimentary to the extent that regular patterns are formed. The odd-composite numbers are only formed from the product of prime factors, each a unique combination.
Looking at the spreadsheet excerpt below, the factors are shown for the non-prime odd-composite numbers as they naturally occur. The order of the factors is with the smallest first.
The second set of columns showing the factors has been rearranged by a spreadsheet formula to show the largest factor for each odd-composite number, first.
