Author Part 2 – Andrew J Frost 12/10/2021 REV TWO PnonP p20
- Post first published:16/11/2020
- Post category:Patterns / Prime Numbers
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Positions of Prime Factor Patterns in Primes and Odd-Composite – Using Gaps – Andy Frost 2020-06-06
Using “My List” as previously described.
To briefly recap on the formation of my list; where n is any integer in the number line (including and starting with 0); then 12n + {1,5,7,11} – sieve of 5 = My List. This list is an infinite Set of numbers which I’m calling my special Set Saf. (This series is illustrated in OEIS A007775).
Using the previously established ending number sequence, 1, 7, 1, 3, 7, 9, 3, 9; I concluded that ending numbers 1 to 7 produce a gap of 6, 7 to 1 produces a gap of 4, 1 to 3 produces a gap of 2, 3 to 7 produces a gap of 4, 7 to 9 produces a gap of 2, 9 to 3 produces a gap of 4, 3 to 9 produces a gap of 6, 9 to 1 is a gap of 2. This originates a gap pattern in the prime/non-prime, odd-composite list, Set Saf as follows: –
Gap Patterns in the Prime/Non-Prime, Odd-Composite list, Set Saf.
The pattern is: – 6, 4, 2, 4, 2, 4, 6, 2, for primes and odd-composites on the number line in accordance with Set Saf . This pattern does not change. This can be checked in the list below which is the beginning of Set Saf : –
1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 217, 221 ……. (red indicates odd-composite non-primes)
The pattern repeats through the entire list continually and infinitely. The irregular spacing between primes is compensated for exactly by a similar but complementary irregular spacing between odd-composite non-primes. The combination of the two sets of numbers from then on forms the rigidly regular repeating pattern within Set Saf.
The pattern indicates the importance of studying the primes together with the odd-composite non-primes as a single Set of numbers, which contains two subsets of, 1) primes, and 2) odd-composite non-primes.
The Odd-Composite Subset
Interestingly, this same Gap pattern 6, 4, 2, 4, 2, 4, 6, 2, exists independently between prime-factors of the Odd-Composite Subset. All the factors are primes equal to or greater than 7.
The Subsets Comprising Factors.
To illustrate this in the Set of numbers Set Saf; for any odd-composite number, choose any prime functioning as a factor of an odd-composite number in this Set. For the sake of ease of illustration I have worked with the first few prime-factors below. That is 7, 11, 13, 17, 19….. etc.
Prime Factors
For prime 7, the first odd-composite that it occurs in is 49, (72 the square), of which 7 is the factor. To find the gap from this odd-composite number of 49, subtract 7 (the previous odd-composite number; i.e. 1 x 7) and the answer is 42, then divide by 7 and the answer is 6.
The next odd-composite is 77; so doing the same again (i.e. subtract the previous odd-composite number 49), the answer is 28, divide by 7 and the result is 4.
The next odd-composite is 91; subtract the previous odd composite 77, the answer is 14, divide by 7 and the result is 2.
The next odd-composite is 119; repeat the process and the result is 4. For 133 the result is 2. For 161 the result is 4. For 161 the result is 6. For 203 the result is 2. Then we go to 217 and the result is 6…. This process can be repeated and infinitely produces the pattern 6,4,2,4,2,4,6,2, | 6,4,2,……
For prime 11, the first odd-composite is 77, of which 11 and 7 are the factors. The pattern is repeating again. So 77 – 11 = 66, and then 66 divided by 11 gives the resultant 6. So 121 – 77 = 44 and 44/11= 4. And 143 – 121 = 22 and 22/11 = 2. And 187 – 143 = 44 and 44/11 = 4. ……etc.
So the formula for this calculation is: –
(prime factor1 x prime factor2 = non-prime odd-composite) – (previous odd-composite)/prime factor = resultant gap number
Prime factors 1&2 are the primes, prime factors, under consideration; previous non-prime odd-composite is literally the previous odd-composite number in the number line list.
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Table excerpts from the Spreadsheet
So for each odd-composite number in Set Saf (as shown in the list above) that is functioning as a factor; the factors on their own follow the same gap pattern 6, 4, 2, 4, 2, 4, 6, 2, …. and this can be seen in the following two illustrated pages from the spreadsheet.
Below, the first page is an extract from the spreadsheet showing the results for primes 7 through to 19, when these primes are acting as factors for the odd-composite numbers in the list, the primes in the list having been (sorted) excluded.
The second page shows an extract from the spreadsheet with the prime numbers (grey rows) included with odd-composite numbers, in the series Set Saf . Also shown are the columns to the left which illustrate the rigid patterns associated with this series. That is “pattern 2”, ending numbers, “pattern 3”, gap numbers. (see the Patterns heading below for greater detail)
To summarise – there are two gap patterns which are the same but are not of the same circumstance, one is inclusive in the list of primes and odd-composite numbers, Set Saf, the other is solely occurring in the Subset of Odd-Composite numbers that exists within within Set Saf.
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Spreadsheet 1 – Factors of Set Saf which is comprised of the Subset of Odd-composite Non-prime numbers.
– The Subset of Odd-composite Non-primes after sorting is shown.
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Spreadsheet 2 – Factors of Set Saf which is comprised of the Subset of Primes and the Subset of Odd-composite Non-primes numbers.
Both these Subsets are shown below with the significant pattern information in the initial columns. The Subset of Prime numbers has grey background.
Patterns
In the image above the following pattern sequences are identified. The 5th column, pattern 3, is the key pattern.
1st column Pattern 4, is showing the ending digit of the starting integer n.
2nd column Pattern 5, is the ending digit of the base number x12.
3rd column Pattern 1, is the prime uplift to raise the product of the integer and x12 base.
4th column Pattern 2, is the ending digit of the prime or odd-composite non-prime produced.
5th column Pattern 3, is my gap sequence across 1) the primes and odd-composite non-primes, and 2) the prime factors of odd-composite numbers in the series.
Pattern 6, shows the repeating sequence of the last two ending digits of the numbers in Set Saf. This series actually has 80 numbers in three Subsets groups, 26, 28, and 26 numbers and repeats thereafter. (See previous posts for explanation).
01 07 11 13 17 19 23 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97
01 03 07 09 13 19 21 27 31 33 37 39 43 49 51 57 61 63 67 69 73 79 81 87 91 93 97 99
03 09 11 17 21 23 27 29 33 39 41 47 51 53 57 59 63 69 71 77 81 83 87 89 93 99
Does the pattern of squares hold true to the gap sequence 6,4,2,4,2,4,6,2…? To see more go to next post no. 21.
“P&NonP – 21 -Positions of Prime Factor Patterns in Odd-Composites pg2”
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hi – just testing 8/3/2022