20211116, 16:49  #1 
Aug 2020
Guarujá  Brasil
2×67 Posts 
An idea for a new class of some numbers.
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 (composited_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) C1 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^2y d_s [x] oblong(d_s [oblong]+1) oblong1 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  Last fiddled with by Charles Kusniec on 20211116 at 16:55 
20211122, 22:39  #2 
"Jeppe"
Jan 2016
Denmark
2^{2}·47 Posts 
For what it's worth, an attempt to format the tables.
Code:
 Table I: Tally Composite C d_s [C] C(d_s [C]+1) C1 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^2y d_s [x] oblong(d_s [oblong]+1) oblong1 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 Last fiddled with by JeppeSN on 20211122 at 23:07 Reason: OEIS links 
20211122, 23:20  #3 
"Jeppe"
Jan 2016
Denmark
10111100_{2} Posts 
You forgot 228 in your first list/table. /JeppeSN

20211123, 03:38  #4 
Aug 2020
Guarujá  Brasil
2×67 Posts 
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.

20211123, 11:27  #5 
Aug 2020
Guarujá  Brasil
206_{8} Posts 
cancel
Last fiddled with by Charles Kusniec on 20211123 at 11:46 Reason: Error in explanation. 
20211123, 11:46  #6 
Aug 2020
Guarujá  Brasil
2×67 Posts 
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:

20211127, 07:53  #7 
Aug 2020
79*6581e4;3*2539e3
1325_{8} Posts 
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: Code:
d_s C 10 93240 12 2383920 16 298378081 18 5133688560 22 73329656400 28 1365328364400 Last fiddled with by bur on 20211127 at 08:41 
20211127, 15:11  #8 
Aug 2020
Guarujá  Brasil
206_{8} Posts 
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? Last fiddled with by Charles Kusniec on 20211127 at 15:30 
20211201, 18:32  #9 
Aug 2020
Guarujá  Brasil
10000110_{2} Posts 
Names for the primes sets.
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 (composite1). 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, (composite1) is not prime. Please evaluate. Thank you, 
20211201, 22:22  #10 
Aug 2020
Guarujá  Brasil
2×67 Posts 
Names for the primes sets. (continuation)
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, Last fiddled with by Charles Kusniec on 20211201 at 22:45 
20211202, 15:50  #11 
Aug 2020
Guarujá  Brasil
2·67 Posts 
New name ideia.
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)) , (composite1) , (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. 
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