Starting an analysis of prime numbers from the composites is something that some number theory books start and then dismiss once the complexities are considered in detail: however in theory, on the basis that composites are relatively straight forward they can be counted and their number subtracted from the total integer count to yield the number of primes. Easy!

A practical difficulty kicks in quickly – well two difficulties actually:

- It’s not as simple as counting each multiple of 2,3,5,7,11 etc..separately (i.e. each prime number or factor) as all composites (other than squares) are the result of multiplying two different primes so can be expressed more than once; thus 6 = 2 x 3 =3 x 2 and so could be counted as a multiple of 2 and a multiple of 3. As another example 60 = 2 x 2 x 3 x 5 could be counted as a multiple of 2, 3 and 5. An adjustment is required.
- The first factor leads to a second related difficulty in that no composites need to be counted against any prime factor P
_{i }where c < P_{i}^{2 }. Thus for prime factor 5 the first composite not already accounted for by prime factors 2 and 3 is 25. The next is 35. Smaller composites such as 10, 15, 20 and 30 are also multiples of 2 or 3 or both. - Once the second factor is taken into consideration we don’t need to worry about adjusting any count for primes themselves.

So let’s make an adjustment for these two difficulties, assuming we start with the smallest prime factor 2 and work upwards:

The number of composites from 2 starts with 4 and continues with every larger even number: as a series these can be expressed as 2n+2 (i.e. where n=1 the term is 4) or counting from the integers as a whole n = I/2 – 1 or (I-2)/2

i.e.

I=4 n=1

I=6 n=2

Thus is we wish to count the number of composites of 2 less than an integer I, we can use I/2 – 1.

We can then count the number of composites from prime factor 3. These start with 9 and continue with every odd multiple of three (the even ones were all accounted for as multiples of 2. The formula 6n+3 applies (i.e. where n=1 the term is 9).

To count the multiples based on integer I we use the formula (I-3)/6.

Combining the count for prime factors 2 and 3 gives a formula of 2(I-3)/3 where the formula produces a whole integer.

Can we continue with this approach? Trouble is, we cannot (easily) as the composites for prime factor 5 (and larger factors) are not uniformly distributed 25,35, 55,65 etc. For easy formulas we need to consider two separate formulas; for prime factors 7 and above we need multiple formulas….the number of which amazingly relate to Euler (see elsewhere).

At this point we need to take a closer examination of the unique composites associated with each prime factor. In the paragraph above the start of the sequence associated with prime factor are listed. As will be seen they occur in pairs and are 30 apart; as will be seen elsewhere these define the well known 6n+1/-1 formula (divide them by 5 if you cannot wait, but one of the tricks is to work out where exactly the series starts).

Things start to get a little “clearer” (maybe complex is a better word initially, but clearer can be used in retrospect) when we look at the unique composites associated with prime factor 7.

With no surprise they start at 49 but then continue in what at first appears a little random.

49, 77, 91, 119, 133, 161, 203, 217, 287, 301, 329, 343, 371, 413, 427

It’s only when you arrange them that you start to see a bit of structure

49,

77, 91, 119, 133, 161, 203, 217, 259

287, 301, 329, 343, 371, 413, 427, 469

putting 49 to one side you can see that the positions of the composites are in rows of 8 with a repeat at intervals of 210. You can keep going for as long as you want; after all the pattern is a result of an interaction of the four prime factors – 2, 3, 5 and 7 – nothing else so this is pretty simple.

If you are interested in the Online Sequence of Integers https://oeis.org/ this sequence is labelled** A063163** and defined as “Composite numbers which in base 7 contain their largest proper factor as a substring”.

We can do this analysis for each subsequent prime factor.

We find that for every prime number p_{i }there is an associated array of composite numbers that consist of the composites derived from p_{i }and larger prime factors but does not include any composites with a prime factor of p_{i }or less.

Each sequence of composites are well defined and can be considered to be arranged in a helix but the start of each subsequent helix is off-set by one prime factor.

The first two “helices” (for prime factors 2 and 3) are trivial in that they each consist of a single line of integers (4,6,8,10 etc and 9,15,21,27 etc). Only with that for prime factor 5 does the helix start to show us some of its features with two lines of integers, but it takes examination of the fourth and fifth before a fuller picture emerges…..and if you have got this far and wondered what this is going to tell you about prime number distribution, just be patient!

There are three dimensions I can give for each of these composite helices:

- there are (P
_{i-1 }– 1)# elements in each turn of the helix - the strands of the helix are p
_{i}# apart - each helix has a line of symmetry at p
_{i }#/2

So here is a starting table (sorry for the limitations of posting a table in wordpress):

p_{i} No of elements in 1st column Gap between columns Line of symmetry

3 1 6

5 2 30 15

7 8 210 105

11 48 2310 1155

13 480 30030 15015

______(P_{i-1 }– 1)# p# p#/2

For those who have explored the prime numbers the above framework may appear familiar. However, arrays of composite numbers, by themselves tell us very little about the primes. You have to move onto the secondary arrays to get an inkling on what is going on…..see “**Secondary arrays**“.

But before moving on we need to examine the composite integers that sit on the above arrays or frameworks. As above it is useful to refer to the Online Encylopedia of Integer Sequences (OEIS) rather than listing long lists of integers: however, there are a number of differences with regards to starting points. I haven’t been too careful on this front so I expect some quality control might be useful.

So, for the composite integers that “sit” on the above frameworks:

OEIS sequence A083140 contains a pretty good summary of the early sequences, but of course the arrays are not acknowledged. The following OEIS sequences make up A083140 but in this context you have to leave off both the first prime number and the second term of the prime squared:

`2 4 6 8 10 12 14 16 18 20 22 24 .... (A005843)`

`3 9 15 21 27 33 39 45 51 57 63 69 .... (A016945)`

`5 25 35 55 65 85 95 115 125 145 155 175 .... (A084967)`

`7 49 77 91 119 133 161 203 217 259 287 301 .... (A084968)`

`11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)`

`13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)`

Thus the repetitive array for prime number 7 starts with 77 and the second column of the array starts with 77 + 210 = 287. Thus the composite array as a whole reads:

77 287 497 707 917 1127 1337 1547

91 301 511 721 931 1141 1351 1561

119 329 539 749 959 1169 1379 1589

133 343 553 763 973 1183 1393 1603

161 371 581 791 1001 1211 1421 1631

203 413 623 833 1043 1253 1463 1673

217 427 637 847 1057 1267 1477 1687

259 469 679 889 1099 1309 1519 1729

As you can see there are 8 rows of integers, and each item in each row is spaced at 210; however, the “line of symmetry” at 105 is not in the middle. One has to rearrange the array to “see” it – two lines need to be brought to the top of the array and offset:

____217 427 637 847 1057 1267 1477 1687

____259 469 679 889 1099 1309 1519 1729

77 287 497 707 917 1127 1337 1547

91 301 511 721 931 1141 1351 1561

————–line of symmetry————————

119 329 539 749 959 1169 1379 1589

133 343 553 763 973 1183 1393 1603

161 371 581 791 1001 1211 1421 1631

203 413 623 833 1043 1253 1463 1673

(Offsetting or steps is something one gets used to with primes, but we’re not on the primes yet!)

So now if one measures the distance of each of the eight rows from a line starting with p#/2 you will see the rows are spaced at equal distances from the mid-line:

-98 (the distance between 217 and 315)

-56 (the distance between 259 and 315)

-28

-14

+14 (=2 x 7)

+28 (= 4 x 7)

+56 (=8 x 7)

+98 (= 14 x 7)

So, an array of equal spaced composite numbers is not too exciting is it? Just keep following – it gets much more interesting: see “**Secondary arrays**”