P&nonP-22 Grouping Patterns of Ending Numbers of Prime Factors in the Odd-Composite Series

Author Part 2 – Andrew J Frost 27/01/2022 PnonP p3

Patterns of Ending Numbers of Prime Factors in the Odd-Composite Series – Using Groupings & Gaps – Andy Frost 2022/01/27

Prime Numbers as Factors

The image below shows the prime numbers as factors occurring in the series my special list Set S_(af), which is defined as primes and odd-composite non-primes.

The Spreadsheet

The spreadsheet is stored in my Mediafire account, and can be accessed for downloading through the link below.

Please look in the folder named “Spreadsheet – P&nonP-page22”. There is also access to all the other spreadsheets posted.

Sometimes there are errors

Note that any errors in the spreadsheets that may be present I can guarantee have occurred due to “data sorting” on the sheet. This occurs where the sheet is not returned to its original status on reversal/undo of the “sort”, which tends to happen with the Calc program.

I’m sure the programmers at LibreOffice Calc would say that it is down to “operator error”, which could well be true! But it has happened too many times for just me to be wrong about this. Can you imagine the amount of work required to reconstitute the sheet to the correct configuration? I now only do “sorts” on spreadsheet copies where the formulae have been removed.

link to spreadsheet

https://www.mediafire.com/file/299ywfx6o7aignp/aP%2526nonP_P-Factors-20220124.ods/file

Above spreadsheet is 27 MB in size.

(To access/load some of my larger spreadsheets (128 MB) you will need to own a computer/laptop with 32 Gig. of memory and a good processor)

Factor Patterns

As stated previously, primes/(prime factors) 2, 3, and 5, are sieved out of my special list. The odd-composite non-primes in the list are only composed from the product of factors of 7 or other odd prime factors greater than 7.

The prime factors occur in a column at the left (field with pink background and blue border) and also as the top row in the spreadsheet extract, where they are coloured red.

The diagonal line marked by the similar cell/field with a pink background and blue border is highlighting the squares of primes occurring as the product of factors.

Factors (r8)

The last column to this area of the spreadsheet headed “Factors R8” shows red numbers in a blue box. These numbers are the ending numbers of the prime factor under consideration, which can be checked against the second column on the left, which is headed “Primes as Factors – Pattern”.

The narrow columns in between the columns showing factors, show the difference value between the factors. This difference is always the “gap pattern” as previously identified and restated below.

Zero

The rows shown with “0”, zero, in all the cells are the rows corresponding with an odd-composite number, which means there cannot be a prime factors occurring, hence the nought. (In this image, see last column which shows the prime numbers listed, with the odd-composite non-primes shown on a yellow background field with magenta border).

Gap Pattern in Relation to Factors

The gap pattern (6,4,2,4,2,4,6,2) applies to the difference between the factors as shown horizontally and vertically of course.

The products of the factors occurring horizontally and vertically, and (as previously explained), the products diagonally shown, which happen to be the squares of the factors; follow the gap pattern.

This data occurs in the previous area in this spreadsheet showing a grey back ground to the data regarding primes against the prime factors of the odd-composite numbers; lines shown with clear backgrounds.
This area occurs just before the area shown in the extracted image above.
The relationship as stated in the previous post (page 2).

This means that there are three separate occurrences of the same gap pattern between these numbers.

Explanation of the Detail

Note – all colours referred to – are the formatting of the spreadsheet – for easier reading of the spreadsheet

Across the spreadsheet extract; there are eight wider columns (pale green backgrounds, header numbers in red from 1 to 8).

There are also further columns showing the factors from 37 to 73 (white backgrounds).

The first column with a pale green background shows the ending number of the product of the prime factor column (second most left column with a pale green background) and the top “factors” row. There are seven more columns with pale green backgrounds demonstrating this relationship.

	*pattern*	6		4		2		4		2		4		6		2		6
			1		2		3		4		5		6		7		8

factors	1	6	7	4	11	2	13	4	17	2	19	4	23	6	29	2	31	6

6	7	6	49	4	77	2	91	4	119	2	133	4	161	6	203	2	217	6
4	11	6	77	4	121	2	143	4	187	2	209	4	253	6	319	2	341	6
2	13	6	91	4	143	2	169	4	221	2	247	4	299	6	377	2	403	6
4	17	6	119	4	187	2	221	4	289	2	323	4	391	6	493	2	527	6
2	19	6	133	4	209	2	247	4	323	2	361	4	437	6	551	2	589	6
4	23	6	161	4	253	2	299	4	391	2	437	4	529	6	667	2	713	6
6	29	6	203	4	319	2	377	4	493	2	551	4	667	6	841	2	899	6
2	31	6	217	4	341	2	403	4	527	2	589	4	713	6	899	2	961	6
	37	6	259	4	407	2	481	4	629	2	703	4	851	6	1073	2	1147	6
	41	6	287	4	451	2	533	4	697	2	779	4	943	6	1189	2	1271	6
	43	6	301	4	473	2	559	4	731	2	817	4	989	6	1247	2	1333	6
	47	6	329	4	517	2	611	4	799	2	893	4	1081	6	1363	2	1457	6
49	0	6	0	4	0	2	0	4	0	2	0	4	0	6	0	2	0	6
	53	6	371	4	583	2	689	4	901	2	1007	4	1219	6	1537	2	1643	6
	59	6	413	4	649	2	767	4	1003	2	1121	4	1357	6	1711	2	1829	6
	61	6	427	4	671	2	793	4	1037	2	1159	4	1403	6	1769	2	1891	6
	67	6	469	4	737	2	871	4	1139	2	1273	4	1541	6	1943	2	2077	6
	71	6	497	4	781	2	923	4	1207	2	1349	4	1633	6	2059	2	2201	6
	73	6	511	4	803	2	949	4	1241	2	1387	4	1679	6	2117	2	2263	6
77	0	6	0	4	0	2	0	4	0	2	0	4	0	6	0	2	0	6
	79	6	553	4	869	2	1027	4	1343	2	1501	4	1817	6	2291	2	2449	6
	83	6	581	4	913	2	1079	4	1411	2	1577	4	1909	6	2407	2	2573	6
	89	6	623	4	979	2	1157	4	1513	2	1691	4	2047	6	2581	2	2759	6
91	0	6	0	4	0	2	0	4	0	2	0	4	0	6	0	2	0	6
	97	6	679	4	1067	2	1261	4	1649	2	1843	4	2231	6	2813	2	3007	6

Factors and Pattern

Then further over the set of eight narrow columns headed “EndNos1 through to EndNos8, corresponds with the above mentioned set of columns. The eight narrow columns show the ending numbers of the numbers in the columns with a wider pale green background.

If we look at the ending numbers in these columns, it is easy to see repeating sequences of numbers in certain patterns.

Repeating Sequences and Patterns in Numbers

Firstly the column headed “Factors R8” identified previously, see above.

It is clear that to demonstrate the existence of the Gap Pattern, we do not need the whole number of the Prime or the Odd-Composite. We only need the Ending Number. The Ending numbers occur in a repeating sequence of eight numbers, “r8”. This sequence is 1,7,1,3,7,9,3,9, clearly shown in the column headed “Factors r8”.

Adjacent to this “Factors r8” column, is the column with a pale yellow background; this shows the Gap Pattern 6,4,2,4,2,4,6,2, which is the difference between these ending numbers regardless of the whole number and continuously repeats.

This then show the relationship between all these odd-composite products of the prime factors as a rigid pattern which does not step out of line.

The rows with 0’s which mark the existence of the odd-composite numbers in my special list also fit into this pattern if you care to work out the product numbers which are not shown. Or you can see these numbers by looking at the spreadsheet where the polynomial formula has been used; that is the series of columns to the right side of this spreadsheet. (Initially shown in post – “PRIMES AND NON-PRIME PATTERNS -11 – Using Equation 2”, on 2 – by Sandor Kristyan).

The reason for omitting these numbers is that they do not relate to primes occurring as factors, but occur as odd-composites complementary to the prime numbers developed in my special list. (Initially shown in post – “Complementarity – Ratio – – – THE RATIO OF NON-PRIMES TO PRIMES).

What’s the Point of All This?

The Ending Numbers of the Ending Number Sequence

The ending numbers that are identified above, in sequence; also occur in different sequences.

The Ending Numbers Columns & Rows

As noted before, there are eight columns identified, but the data repetition is not limited to eight columns. These eight are a sample. They are headed, EndNos1 through to EndNos8. They contain vertically their own “r8” repeating sequence of eight numbers.

.	Factors EndNos	G	End Nos1	End Nos2	End Nos3	End Nos4	End Nos5	End Nos6	End Nos7	End Nos8	End Nos9	repeat sequ.	Group
			r8	r8	r8	r8	r8	r8	r8	r8
			1	2	3	4	5	6	7	8

1	1		7	1	3	7	9	3	9	1	7	1379	c
2	7	6	9	7	1	9	3	1	3	7	9	1379	d
3	1	4	7	1	3	7	9	3	9	1	7	1379	c
4	3	2	1	3	9	1	7	9	7	3	1	9731	a
5	7	4	9	7	1	9	3	1	3	7	9	1379	d
6	9	2	3	9	7	3	1	7	1	9	3	9731	b
7	3	4	1	3	9	1	7	9	7	3	1	9731	a
8	9	6	3	9	7	3	1	7	1	9	3	9731	d

	1	2	7	1	3	7	9	3	9	1	7	1379	c
	7	6	9	7	1	9	3	1	3	7	9	1379	d
	1	4	7	1	3	7	9	3	9	1	7	1379	c
			1379	1379	9731	1379	9731	9731	9731	1379	1379

Ending Numbers Sequences

The above table shows the first eight rows, plus three rows, plus one; also the first eight columns, plus one, plus another one, plus groups; as an extract from the spreadsheet of this area.

Vertical Sequences

Patterns such as the sequence of four ending numbers shown in the table, i.e. 1,3,7,9 and the reverse 9,7,3,1, occur in this table and therefore the the numbers that are the products of the prime factors. These numbers repeat infinitely in the plane of a typically x,y field, that is vertically and horizontally. However to see this it was necessary to add a few more columns and rows than just eight.

Evolving the Repeating Horizontal Sequence

There is a more complicated pattern occurring however. This concerns the sequences of eight numbers shown vertically and horizontally; but the consideration is predominately on the horizontal numbers.

Horizontal Sequences

For every prime factor there is an eight digit code sequence which relates to that prime factor’s ending number. This sequence is not unique, it is a repeating pattern.

This number is set by the “gap sequence” shown in the table vertically in the column headed “G” as a value of difference between the rows. But it can also be worked out for the column.

However the “gap sequence” also applies across the table. This has been explained before in “Formula Illustrating the Gap Pattern Between Factors” in Page Two . Simply put, the relationship is a product; the gap sequence number, times the prime factor, added to the previous odd-composite number.

Grouping of Prime Factor (Ending Number) Patterns

Horizontal Repeating Sequence

When the prime factor is: –

7, the sequence is – 9, 7, 1, 9, 3, 1, 3, 7, corresponding to odd-composite prime-factor products: – 49, 77, 91, 119, 133, 161, 203, 217 – group “d”.
11, ending number 1, the sequence is – 7, 1, 3, 7, 9, 3, ,9, 1, corresponding to odd-composite prime-factor products: – 77, 121, 143, 187, 209, 253, 319, 341 – group “c”.
13, ending number 3, the sequence is – 1, 3, 9, 1, 7, 9, 7, 3, corresponding to odd-composite prime-factor products: – 91, 143, 169, 221, 247, 299, 377, 403 – group “a”
17, ending number 7, the sequence is – 9, 7, 1, 9, 3, 1, 3, 7, corresponding to odd-composite prime-factor products: – 119, 187, 221, 289, 323, 391, 493, 527 – group “d”
19, ending number 9, the sequence is – 3, 9, 7, 3, 1, 7, 1, 9, corresponding to odd-composite prime-factor products: – 133, 209, 247, 323, 323, 361, 437, 551 – group “b”
23, ending number 3, the sequence is – 1, 3, 9, 1, 7, 9, 7, 3, corresponding to odd-composite prime-factor products: – 161, 253, 299, 391, 437, 529, 667, 713 – group “a”
29, ending number 9, the sequence is – 3, 9, 7, 3, 1, 7, 1, 9, corresponding to odd-composite prime-factor products: – 203, 319, 377, 493, 551, 667, 841, 899 – group “b”
31, ending number 1, the sequence is – 7, 1, 3, 7, 9, 3, ,9, 1, corresponding to odd-composite prime-factor products: – 217, 341, 403, 527, 589, 713, 899, 961 – group “c”.

There are only four patterns, as can be seen above; these patterns repeat endlessly through the primes as prime-factors in my special list Set S_(af). The patterns are identified with a group letter of “a”, “b”, “c”, or “d”.

Grouping

Grouping of the ending numbers, means grouping of primes and odd-composite non-primes in to just these four groups and shows this fact.

Structure

Above is demonstrated a network of patterns and groups that occur within this Set of numbers. This looks like a structure upon which the primes and odd-composite non-primes are built.

end