Author – Andrew J Frost 26/09/2021 REV TWO PnonP p18
OBSERVATION 18 – Summaries So Far
Today 08-10-2021
The prime P and non-prime infinite Set S(af) (see Appendix 1)
(the red highlighted numbers are non-primes)
1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89,91,97,101,103,107,109,113,119,121,127,131,133,137,139, 143,149,151,157,161,163,167,169,173,179,181,187,191,193,197,199,203,209,211,217,221,223,227,229,233,239,241,247,251,253, 257,259,263,269,271,277,281,283,287,289,293,299,301,307,311,313,317,319,323,329,331,337,341,343,347,349,353,359,361,367, 371,373,377,379,383,389,391,397,401,403,407,409,413,419,421,427,431,433,437,439,443,449,451,457,461,463,467,469,473,479, 481,487,491,493,497,499,503,509,511,517,521,523,527,529,533,539,541,547,551,553,557,559,563,569,571,577,581,583,587,589, 593,599,601,607,611,613,617,619,623,629,631….
The gaps sequence 64242462
The gaps sequence that occurs in the prime P and non-prime infinite Set S(af), which is a rigid immutable repeating sequence (conjecture), and occurs in apparently many places but not necessarily linked. This pattern is found as a source to other sequences of numbers, these sequences occurring in many places to do with primes and odd-composite non-primes.
6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2…..
The integer series required to originate the prime and non-prime odd-composite numbers of the S(af) infinite set. (where 2n+1 = P or nonP)
3,5,6,8,9,11,14,15,18,20,21,23,24,26,29,30,33,35,36,38,39,41,44,45,48,50,51,53,54,56,59,60,63,65,66,68,69,71,74,75,78,80,81,83, 84,86,89,90,93,95,96,98,99,101,104,105,108,110,111,113,114,116,119,120,123,125,126,128,129,131,134,135,138,140,141,143, 144,146,149,150,153,155,156,158,159,161,164,165,168,170,171,173,174,176,179,180,183,185,186,188,189,191,194,195,198,200,201,203,204,206,209,210,213,215,216,218,219,221,224,225,228,230,231,233,234,236,239,240,243,245,246,248,249,251,254,255,258,260,261…
Above is the sequence of integers (introw & intcol) that generates the data for the spreadsheet of My Set S(af) (Equation 2); the resultants include the primes and non-primes, prime squares and all other data generated. This has been extracted from my spreadsheet.
The prime integer series required to originate only the primes in the S(af) infinite set. (The highlighted red numbers above are now omitted). (OEIS A005097)
3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36,39,41,44,48,50,51,53,54,56,63,65,68,69,74,75,78,81,83,86,89,90,95,96,98,99,105,111,113,114,116,119,120,125,128,131,134,135,138,140,141,146,153,155,156,158,165,168,173,174,176,179,183,186,189,191,194,198,200,204,209, 210,215,216,219,221,224,228,230,231,233,239,243,245,249,251,254,260,261….
Integers generating only the odd-composite numbers in the prime and non-prime Set S(af) as highlighted above (and in the spreadsheet using Eq2).
24,38,45,59,60,66,71,80,84,93,101,104,108,110,123,126,129,143,144,149,150,159,161,164,170,171,180,185,188,195,201,203,206,213,218,225,234,236,240,246,248,255,258…
The gaps sequence 21212313 that occurs in the integer series within infinite set S(af) which is a rigid immutable sequence (conjecture), and linked to the gaps sequence 42424626.
3,2,1,2,1,2,3,1,3,2,1,2,1,2,3,1,3,2,1,2,1,2,3,1,3,2,1,2,1,2,3,1,3,2,1,2,1,2,3,1,3,2,1,2,1,2,3,1,3,2,1,2,1,2,3,1,3,2,1,2,1,2,3,1…..
The gaps between of the terms in the prime only integer series above, required to originate the prime numbers of the S(af) infinite set, which is an irregular and non-repeating infinite series.
3,2,1,2,1,2,3,1,3,2,1,2,3,3,1,3,2,1,3,2,3,4,2,1,2,1,2,7,2,3,1,5,1,3,3,2,3,3,1,5,1,2,1,6,6,2,1,2,3,1,5,3,3,3,1,3,2,1,5,7,2,1,2,7,3,5,1,2,3,4,3,3,2,3,4,2,4,5,1,5,1,3,2,3,4,2,1,2,6,4,2,4,2,3,6,1…
(1, 1,) 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 3, 1, 3, 2, 1, 3, 2, 3, 4, 2, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 3, 3, 2, 3, 3, 1, 5, 1, 2, 1, 6, 6, 2, 1, 2, 3, 1, 5, 3, 3, 3, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 3, 3, 2, 3, 4, 2, 4, 5, 1, 5, 1, 3, 2, 3, 4, 2, 1, 2, 6, 4, 2, 4, 2, 3, 6, 1, 9, (OEIS A028334)
See the table representation above of the prime gaps and non-prime gaps required to generate a list of primes only.
End of Part One
