An idea for a new class of some numbers.

[Charles Kusniec]

Let us define d_s [composite] is the number of divisors in the first sequence of consecutive divisors of a composite number. Consequently, it is only possible to have the closest prime numbers in the form of (composite±(d_s [composite]+1)) or (composite±1).

Let us take the example of the number 12:
The composite 12 has six divisors: 1,2,3,4,6,12.
The first sequence of consecutive divisors of 12 has four divisors: 1,2,3,4.
Then, d_s [12]=4.
Because of the four consecutive divisors, then 12±2, 12±3, and 12±4 cannot be prime.
So, the closest primes of 12 may occur at (composite±1)=(12±1) and at (composite±(4+1))=(12±5).
Indeed, the composite 12 has two primes in the form of (12±1): primes 11 and 13 and two primes in the form of (12±5): primes 7 and 17.

This is true for any composite. This is because, if d_s value is the last divisor value of the first sequence of consecutive divisors of a composite, then all consecutive numbers between (composite-d_s ) and (composite+d_s ) will also have at least one of the divisors of the first sequence of divisors of the composite.

This can avoid some waste of time searching big prime numbers.
For example, composite 60 has d_s (60)=6.
In this case, to find the primes 53, 59, 61, 67 we just need to apply the formulas (60±1) and (60±d_s (60)±1). The HCN 60 has d_s long enough that we can ignore oblong 56 and square 64. Or we can say, the HCN 60 has d_s long enough that we can ignore (56±1) and (64±1) search as possible primes.

We use a similar idea to find records for prime numbers of Mersenne type. We can extend to primes in the form of (2^n±(d_s+1)).

The sequence {12, 60, 108, 312, 600, 1092, 1428, 1488, 1620, 1872, 2340, 2688, 3540, …} is the first 13 composites that generate primes in all 4 forms (composite±(d_s [composite]+1)) and (composite±1):

-----------------------
Table I:
Tally Composite C d_s [C] C-(d_s [C]+1) C-1 C+1 C+(d_s [C]+1)
1 12 4 7 11 13 17
2 60 6 53 59 61 67
3 108 4 103 107 109 113
4 312 4 307 311 313 317
5 600 6 593 599 601 607
6 1092 4 1087 1091 1093 1097
7 1428 4 1423 1427 1429 1433
8 1488 4 1483 1487 1489 1493
9 1620 6 1613 1619 1621 1627
10 1872 4 1867 1871 1873 1877
11 2340 6 2333 2339 2341 2347
12 2688 4 2683 2687 2689 2693
13 3540 6 3533 3539 3541 3547
--------------------

We also can introduce some variations: instead of 4 primes, 3 primes, etc.
Because between 2 square numbers always exist at least 2 primes numbers with an oblong number between them, then we can conjecture that the oblong numbers are the densest polynomial generators of prime numbers in the form of (composite±(d_s [composite]+1)) and (composite±1).

The sequence {12, 600, 3540, 35532, 245520, 17110632, …} is the first 6 oblong numbers that generate primes in all 4 forms (oblong±(d_s [oblong]+1)) and (oblong±1).

------------------------
Table II:
Tally y x=y^2-y d_s [x] oblong-(d_s [oblong]+1) oblong-1 oblong+1 oblong+(d_s [oblong]+1)
1 4 12 4 7 11 13 17
2 25 600 6 593 599 601 607
3 60 3540 6 3533 3539 3541 3547
4 189 35532 4 35527 35531 35533 35537
5 496 245520 6 245513 245519 245521 245527
6 4137 17110632 4 17110627 17110631 17110633 17110637
------------------------

[JeppeSN]

For what it's worth, an attempt to format the tables.

-----------------------
Table I:
Tally   Composite C   d_s [C]   C-(d_s [C]+1)   C-1    C+1    C+(d_s [C]+1)
1       12            4         7               11     13     17
2       60            6         53              59     61     67
3       108           4         103             107    109    113
4       312           4         307             311    313    317
5       600           6         593             599    601    607
6       1092          4         1087            1091   1093   1097
7       1428          4         1423            1427   1429   1433
8       1488          4         1483            1487   1489   1493
9       1620          6         1613            1619   1621   1627
10      1872          4         1867            1871   1873   1877
11      2340          6         2333            2339   2341   2347
12      2688          4         2683            2687   2689   2693
13      3540          6         3533            3539   3541   3547
--------------------


------------------------
Table II:
Tally   y      x=y^2-y    d_s [x]   oblong-(d_s [oblong]+1)   oblong-1   oblong+1   oblong+(d_s [oblong]+1)
1       4      12         4         7                         11         13         17
2       25     600        6         593                       599        601        607
3       60     3540       6         3533                      3539       3541       3547
4       189    35532      4         35527                     35531      35533      35537
5       496    245520     6         245513                    245519     245521     245527
6       4137   17110632   4         17110627                  17110631   17110633   17110637
------------------------

What is called here d_s [n] is the same as A055874(n); and also d_s [n]+1 is A007978(n).

/JeppeSN

[JeppeSN]

You forgot 228 in your first list/table. /JeppeSN

[Charles Kusniec]

Dear JeppeSN, I don't know why, but yes I skipped 228 in Table 1. Thank you for this and your other comments, especially the tip on the sequence A055874.

[Charles Kusniec]

cancel

[Charles Kusniec]

There are many other interesting things to study in the direction of this class of prime numbers. For example: (1) in table 1, all composites are distanced by a multiple of 12 units; (2) the same is true for twin primes; (3) but, with the columns C +- (d_s[C] + 1) only occurs when d_s does not vary. See table 1 with their differences and their values divided by 12:

[bur]

Interesting idea, I tried to find an example for d_s > 6 but there were none up to d_s = 102. I only tested the smallest possible C with that d_s.

I did the search manually though as I was too lazy to come up with an efficient way to do it with a script. You'd need to determine whether to multiply C simply by the next prime or one of the smaller primes. I couldn't find a quick way to check if the next divisor is a power of a single distinct prime factor or not.

What might make more sense is looking not only at the smallest number for a given d_s, but at all possible ones. For any d_s that is p^n - 1 there cannot be any primes since the numbers will always be divisible by p. So next after d_s = 6 would be d_s = 10. These are the smallest composites for that d_s that will yield all primes. You might call them Kusniec numbers:

d_s C
10  93240
12  2383920
16  298378081
18  5133688560
22  73329656400
28  1365328364400

Is there a Kusniec number for every d_s?

[Charles Kusniec]

Dear Bur, thank you very much for your comments and suggestions.
I would not expect continuous behavior for all d_s. The reason is that all divisors d_s will always be less than sqrt(n). This means that somehow the quadratic function will "modulate" the limit of d_s. So I don't think it can be continuous.
By the way, did you have a chance to check about oblongs as composites?

[Charles Kusniec]

Regarding the names I would like to propose for your evaluation:
1. Let's call "quartet primes" a set of four primes in the form of (composite±(d_s [composite]+1)) and (composite±1).
2. Let us call "lower trio twin primes" a set of three primes in the form (composite-(d_s [composite]+1)) and (composite±1). In this case, (composite+(d_s [composite]+1)) is not prime.
3. Let's call "upper trio twin primes" a set of three primes in the form of (composite+(d_s [composite]+1)) and (composite±1). In this case, (composite-(d_s [composite]+1)) is not prime.
4. Let's call "lower trio primes" a set of three primes in the form (composite±(d_s [composite]+1)) and (composite-1). In this case, (composite+1) is not prime.
5. Let's call "upper trio primes" a set of three primes in the form (composite±(d_s [composite]+1)) and (composite+1). In this case, (composite-1) is not prime.
Please evaluate.
Thank you,

[Charles Kusniec]

Regarding the names I missed for your evaluation:
6. Let's call "duet primes" a set of two primes in the form of (composite±(d_s [composite]+1)). In this case, (composite±1) are not primes.
Please evaluate.
Thank you,

[Charles Kusniec]

I think it's important to properly name these sets of prime numbers. It helps when we make comparisons and detect some properties of prime numbers.
So, without wanting to be repetitive or abusive, I'm going to give another idea of how to name these sets of prime numbers in a complete and mnemonic way.

Each composite always generates a sequence of 4 elements in the form of:
( (composite-(d_s [composite]+1)) , (composite-1) , (composite+1) , (composite+(d_s [composite]+1)) ).
So, we will assign the letter P if element is a prime number, and the letter C if element is a composite number.

Example:
1. Composite 12 generates a (PPPP) prime set.
2. Composite 18 generates a (CPPC) prime set (only twin primes).
3. Composite 20 generates a (PPCP) prime set.
4. Composite 26 generates a (PCCP) prime set (only “duet” primes).
And so on.