2022-09-30, 14:09 | #1 |
Aug 2020
Guarujá - Brasil
2·5·13 Posts |
C001202 The SOM numbers
C001202 The SOM numbers
We will call C001202 the SOM numbers the https://oeis.org/A000290 square numbers interleave with the https://oeis.org/A002378 oblong numbers interleave with the https://oeis.org/A005563 (square–1) numbers. The name SOM numbers comes from S for square, O for oblong, and M for square minus 1. It is the https://oeis.org/A000290 square numbers interleave with https://oeis.org/A035106. It is the https://oeis.org/A002378 oblong numbers interleave with https://oeis.org/A135276. It is the https://oeis.org/A005563 (square–1) numbers interleave with https://oeis.org/A002620. The positive SOM numbers are the OEIS sequence https://oeis.org/A006446 = {1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 20, 24, 25, 30, 35, 36, 42, 48, 49, ...}. The first 3 positive SOM numbers are {1,2,3}. This sequence has five 0´s: one of the squares, two of the oblongs, and two of the (square minus 1) numbers. To consider the 0 elements of the sequence, we must describe the sequence as \(C001202={…, 16, 15, 12, 9, 8, 6, 4, 3, 2, 1, 0, 0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 9, 12, 15, 16, ...}\). Except for the elements \(0,1,2,3\), all other SOM numbers are composites. During our studies, we found that these 3 types of numbers follow the entire number line in a repetitive and uniform quadratic manner with no omissions or repetitions. Thus, these numbers are the reference points for our analysis of the number line. The SOM numbers work like a scale on the number line. They work like the marks on a ruler or tape measure. The prime numbers are distributed between the SOM numbers according to https://oeis.org/A307508 and https://oeis.org/A334163. See the triangular distribution of primes between squares and oblongs in C000885 The Athanasii Kircheri triangle filled with non-negative integers. The SOM numbers occupy the two side edges and the central axis of the Kircheri triangle: Figure C001202 The SOM numbers https://oeis.org/A006446 in the Kircheri triangle. Because there is no integer number between a (square–1) number and a square number, we can conclude that:
The SOM numbers produce the 3 main quadratic CG - composite generators in the FMT – full multiplication table. Because of the geometry of hyperbolas in FMT, from the three initial \(1:1\) diagonals square \(y(y±0)\), oblong \(y(y±1)\) and square–1 \(y(y±2)\), all other quadratic sequences will always have one or more repeated element equal to one or more elements from some previous \(1:1\) diagonals. The three main quadratic CG sequences are:
They are the main CGs because, it is geometrically impossible for any hyperbola as \(HL[x,y]=xy\) simultaneously cross the dots with integer coordinates between all three sequences. As we move away from the square numbers CG \(1:1\) line, new CGs will always have some elements that appeared in former CGs. Any other quadratic \(CG[y]=y(y±b_d)\) will have a repeated composite from a former \(CG[y]=y(y±(b_d-n))\). For example, number \(4\) in \(y(y±3)\) is repeated composite from \(y(y±0)\). Number \(6\) in \(y(y±5)\) is repeated composite from \(y(y±1)\). And so on. Any composite number \(C\) in \(y(y±(C-1))\) is a repeated composite from some former CG \(y(y±(C-1-n))\). The first repetition of TMT - triangular multiplication table elements occurs with number \(4\) at \(y(y±3)\), which is the (oblong–2) numbers. Number 4 is the lowest composite to repeat. It belongs to https://oeis.org/A000290 the square numbers \(y(y±0)\) and https://oeis.org/A028552 the (oblong–2) numbers \(y(y±3)\). The sequence https://oeis.org/A334454 shows an example of this application. Last fiddled with by Charles Kusniec on 2022-09-30 at 14:16 Reason: Just format. |
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