C001202 The SOM numbers

[Charles Kusniec]

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.