Ripple Numbers

In this blog post, I am coining the term "Ripple Numbers", a new integer sequence, which has got some very interesting properties.

The definition goes as follows:

For any positive integer s (call it seed), when it is expressed as the product of two numbers mxn (m, n > 0), the base ripple number (call it b) will be equal to 2(m+n+2).

The subsequent ripples take the form b + 8r, where the values of r are 1, 2, 3, .. and so on.

A ripple number can be thought of as a wave which covers a seed number fully in all directions.

For example take number 6. It can be expressed as the product of two numbers 2x3.

So the ripple numbers for 6 can be computed as shown below.

But if we take another factorisation of 6, which is 1x6, then the ripples look like the following.

So, it can be observed that, based on the factorisation of the seed number, the ripples will have a very different representation for the same seed value.

When one of the multiples of the seed factors is 1, then the base ripple expression condenses into more compact form as below.

Let us suppose for the seed s=mxn, the value of m=1.

Then the base ripple expression 2(m+n+2) becomes 2(n+3).

Also the sum of the seed and base ripple becomes 1xn + 2(n+3) = 3n+6.

These two expressions are very useful, and the tables below are the listing of ripple number sequences and their consecutive sums for the first 6 seeds as per the compact formulae mentioned above.

Ripples:

{1, 08, 16, 24, 32, 40, 48, 56, 64, 72, 80}

{2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82}

{3, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84}

{4, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86}

{5, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88}

{6, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90}

Sum Series:

01 -> {09, 25, 49, 81, 121, 169, 225, 289, 361, 441}

02 -> {12, 30, 56, 90, 132, 182, 240, 306, 380, 462}

03 -> {15, 35, 63, 99, 143, 195, 255, 323, 399, 483}

04 -> {18, 40, 70, 108, 154, 208, 270, 340, 418, 504}

05 -> {21, 45, 77, 117, 165, 221, 285, 357, 437, 525}

06 -> {24, 50, 84, 126, 176, 234, 300, 374, 456, 546}

It can be observed in the sum series that, it is forming squares of odd numbers starting with 3, 5, 7, ... when the seed value is 1. 

And also it can be observed that the whole series is forming the odd number tables 3, 5, 7, ... vertically.

So, the sum series has got the special property of generating all odd number tables, without any need for multiplication, which is very useful in generating prime number sieves.

Formulation of the Sum Series:

The reason why the sum series is forming odd number tables is because, the sum of the consecutive ripple numbers along with their seed, forms the composite numbers of the form (m+x)*(n+x) where x is the set of even numbers starting from 2.

The proof goes as follows:

For a seed of the form mxn, it is known that the base ripple is equal to 2(m+n+2).

Adding the seed and the base ripple, yields the expression:

mn + 2(m+n+2) = mn+ 2m + 2n + 4

= m(n+2) + 2(n+2) = (m+2) * (n+2)

By adding the second ripple to the above expression, it becomes:

mn + (2m + 2n + 4) + [(2m + 2n + 4) + 8] = mn + 4m + 4n + 16
= m(n+4) + 4(n+4) = (m+4) * (n+4)

So, by the theory of induction, the next ripple yields the sum (m+6)*(n+6)... so on.

Ripple Sieve for Prime Number Generation:

As said earlier, the ripple sum series (using the compact formulae) can be used for sieving primes, by generating the series for the odd values of seed numbers 1, 3, 5, ... etc., because we all know that primes end with the digits 1,3,7, and 9 only, and exclude any multiples of 2.

The computational efficiency of this sieve exactly matches with the Sieve of Eratosthenes, but the limit check goes far beyond to (n-6)/3, against square root of (n) of Eratosthenes sieve, making it not very suitable for sieving.

Notes:

While the Ripple Sieve is not as efficient as Eratosthenes, it almost matches in the time complexity with another very elegant, but very less known prime sieving algorithm developed by the Indian Mathematician, S. P. Sundaram. Please follow the links below to know more about it.

https://en.wikipedia.org/wiki/Sieve_of_Sundaram

https://luckytoilet.wordpress.com/2010/04/18/the-sieve-of-sundaram/

The ripple number sequence {1, 08, 16, 24, 32, 40, 48, 56, 64, 72, 80, ...} matches with the co-ordination sequence of Root Lattice B2, but it is out of scope for the discussion of this blog. Following is the wiki link to know more about root systems and lattices.

https://en.wikipedia.org/wiki/Root_system