 mersenneforum.org C001202 The SOM numbers
 Register FAQ Search Today's Posts Mark Forums Read 2022-09-30, 14:09 #1 Charles Kusniec   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:Between two consecutive https://oeis.org/A000290 square numbers, there is one oblong number and one (square–1) number. Between two consecutive https://oeis.org/A005563 (square–1) numbers, there is one oblong number and one square number. Between two consecutive https://oeis.org/A002378 oblong numbers, there is one (square–1) number and one square number. It is possible to interleave the oblong numbers and square numbers. The sequence in the OEIS is https://oeis.org/A002620. It is possible to interleave the oblong numbers and (square–1) numbers. The sequence in the OEIS are almost https://oeis.org/A035106 and/or https://oeis.org/A198442. It is possible to interleave the square numbers and (square–1) numbers. The sequence in the OEIS is https://oeis.org/A135276. 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:$$y(y±0)$$ the square numbers, https://oeis.org/A000290, has number 1. $$y(y±1)$$ the oblong numbers, https://oeis.org/A002378, has number 2. $$y(y±2)$$ the square-1 numbers, https://oeis.org/A005563 has number 3. 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. Attached Thumbnails Last fiddled with by Charles Kusniec on 2022-09-30 at 14:16 Reason: Just format.  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