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 2021-12-05, 21:04 #12 Charles Kusniec     Aug 2020 Brasil 25 Posts Thanks to the great help of Michel Marcus, OEIS Editor, he generated all the possible results of the sequences in the form of ( (composite-(d_s [composite]+1)) (composite-1) , (composite+1) , (composite+(d_s [composite]+1)) ). First he did up to 10^5 and then up to 10^6. When we put this data into the SAS program, we see that the highest frequency of the prime number generator is CCCP followed closely by PCCC. Since he was the first to see this and sent me the file with the data named "Soviet", I think that these sequences of composites that generate primes in the above form should be called "soviet composite system" or something similar. See that this "soviet system" makes half of the composites up to 10^6 generate at least one prime number. He also asked me why I am directing the soviet system only to composites. I answered that all odd prime numbers have d_s=1. So the most that a prime number can generate is another prime number which is the complement where the two will be twin primes. The purpose of creating this "soviet system" is to find a very large prime number from a composite number created in the size we wish. That is, we can create a composite number (not necessarily a factorial) but that has a very long d_s and that is above a certain number of digits and, from it, or some close ones, use the soviet system to find at least one prime number that has as many digits as we want. This is only possible with the composites. And I conjecture that the densest composite in the form of a quadratic (or any polynomial form) must be an oblong number. This is because: (a) the oblong numbers greater than 0 are equidistant between any square and the next (square minus one) number, (b) all squares>1, oblongs>2 and square minus one>3 are always composite, and (c) for sure there are at least 2 distinct primes between 2 consecutive square numbers: (1) one between the lower square and the following oblong https://oeis.org/A307508 and (2) another between the oblong and the higher square https://oeis.org/A334163. Attached Thumbnails     Last fiddled with by Charles Kusniec on 2021-12-05 at 21:56
 2021-12-06, 22:30 #13 Charles Kusniec     Aug 2020 Brasil 408 Posts The prime 5 is the only prime in PCCP sequence. The PCCP sequence is very similar to https://oeis.org/A225461. There is a note in A225461: "Heuristically, A001223(n)/2 should be prime with probability ~ 1/log log n, so (prime(n) + prime(n+1))/2 is in the sequence with probability ~ 1/log log n. - Robert Israel, Nov 30 2015". So, may we expect similar probability in PCCP?
 2021-12-08, 13:40 #14 Charles Kusniec     Aug 2020 Brasil 25 Posts Similar to Brun theorem at https://en.wikipedia.org/wiki/Brun%27s_theorem, we can calculate various constants for all possible duets, trios and quartets of primes in the form of "soviets". Would anyone be willing to collaborate with these calculations by making the programs and posting the results here? Thank you. Last fiddled with by Dr Sardonicus on 2021-12-08 at 13:52 Reason: fignix topsy
 2021-12-12, 14:35 #15 Charles Kusniec     Aug 2020 Brasil 1000002 Posts Important features and new sequences We can express uniquely all integers as the product of their two complementary central divisors. We call these two divisors as the central pair of complementary divisors. These two central divisors are only equal in the square numbers. Let us call $$d_{c1}$$ the smallest central divisor. Its values are the values of the sequence https://oeis.org/A033676. Let us call $$d_{c2}$$ the largest central divisor. Its values are the values of the sequence https://oeis.org/A033677. Let us call $$b_m$$ the minimum difference between the two complementary divisors $$b_m=d_{c2}-d_{c1}$$. Its values are the values of the sequence https://oeis.org/A056737. Let us call $$d_s$$ the number of divisors in the first sequence of consecutive divisors of a composite number. Its values are the values of the sequence https://oeis.org/A055874. Let us call $$d_r$$ the complementary divisor of the number of divisors in the first sequence of consecutive divisors of a composite number. Its first 60 positive elements are AAAAA1={1, 1, 3, 2, 5, 2, 7, 4, 9, 5, 11, 3, 13, 7, 15, 8, 17, 6, 19, 10, 21, 11, 23, 6, 25, 13, 27, 14, 29, 10, 31, 16, 33, 17, 35, 9, 37, 19, 39, 20, 41, 14, 43, 22, 45, 23, 47, 12, 49, 25, 51, 26, 53, 18, 55, 28, 57, 29, 59, 10}. Then, $$integer=n=d_{c1}*d_{c2}=d_s*d_r$$. The integers that have $$d_s=d_{c1}$$ and $$d_r=d_{c2}$$ form a sequence. Its first 60 positive elements are AAAAA2={1, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 18, 19, 22, 23, 24, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 60, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151}. The AAAAA2 sequence have all odd primes from the sequence https://oeis.org/A065091. All these odd primes elements, including element 1, have $$d_s=d_{c1}=1$$. The other elements that are not prime numbers in the sequence AAAAA2 form the sequence AAAAA3. Its first 60 positive elements are AAAAA3={1, 4, 8, 10, 14, 18, 22, 24, 26, 34, 38, 46, 58, 60, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526} Excluding the elements AAAAA4={1, 8, 18, 24, 60}, all other elements of AAAAA3 are 2*prime. All these 2*primes elements, including element 8, have $$d_s=d_{c1}=2$$. Element 18 has $$d_s=d_{c1}=3$$. Element 24 has $$d_s=d_{c1}=4$$. Element 60 has $$d_s=d_{c1}=6$$. It is not possible exist $$d_s=d_{c1}=5$$, because automatically is divisible by 6=2*3. Let us call $$b_r$$ the difference between the two complementary divisors $$b_r=d_r-d_s$$. Its first 60 positive elements are AAAAA5={0, -1, 2, 0, 4, -1, 6, 2, 8, 3, 10, -1, 12, 5, 14, 6, 16, 3, 18, 8, 20, 9, 22, 2, 24, 11, 26, 12, 28, 7, 30, 14, 32, 15, 34, 5, 36, 17, 38, 18, 40, 11, 42, 20, 44, 21, 46, 8, 48, 23, 50, 24, 52, 15, 54, 26, 56, 27, 58, 4}. All records in odd integers. The 3 oblong elements AAAAA6={2,6,12}={1*2,2*3,3*4} are the only elements with $$d_s>d_r$$ with $$d_r-d_s=-1$$. The 2 square elements AAAAA7={1,4}={1*1,2*2} are the only elements with $$d_s=d_r.$$ Last fiddled with by Charles Kusniec on 2021-12-12 at 15:16
 2021-12-12, 23:02 #16 Charles Kusniec     Aug 2020 Brasil 25 Posts I noticed now: there is an error in the 2 definitions. The correct is: Let us call $$d_s$$ the number of divisors in the first sequence of consecutive divisors of an integer number. Its values are the values of the sequence https://oeis.org/A055874. Let us call $$d_r$$ the complementary divisor of the number of divisors in the first sequence of consecutive divisors of an integer number.
 2021-12-22, 13:00 #17 Charles Kusniec     Aug 2020 Brasil 25 Posts The integer numbers in the form of CCCC. The integers a(n) in the form of CCCC are those that produce 4 composites in the form of $$(a(n)+-(d_s+1))$$ and $$(a(n)+-1)$$. They form the sequence AAAAAA The primes a(n) that produce composite numbers in the forms of $$(a(n)+-(d_s+1))$$ and $$(a(n)+-1)$$. = {23, 37, 47, 53, 67, 79, 83, 89, 93, 97, 113, 117, 118, 119, 121, 122, 123, 127, 131, 143, 145, 157, 163, 167, 173, 185, 186, 187, 203, 205, 206, 207, 211, 215, 217, 218, 219, 223, 233, 245, 246, 247, 251, 257, 263, 277, 287, 289, 293, 297, 298, 299, 301, 302, 303, 307, 317, 321, 322, 323,...}. All prime numbers of this AAAAAA sequence are single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime. They belong to the sequences https://oeis.org/A007510 and https://oeis.org/A134797. All composite numbers from AAAAAA form the sequence BBBBBB The composites a(n) that produce composite numbers in the forms of $$(a(n)+-(d_s+1))$$ and $$(a(n)+-1)$$ ={93, 117, 118, 119, 121, 122, 123, 143, 145, 185, 186, 187, 203, 205, 206, 207, 215, 217, 218, 219, 245, 246, 247, 287, 289, 297, 298, 299, 301, 302, 303, 321, 322, 323, 324, 325, 326, 327, 341, 342, ...}.

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