The box ■ operation
Hello SeqFans,
Let's define (a ■ b) = c with an example:
12951
■ 2019
--------
= 10948
We align a and b on the right and make the absolute differences of the vertically disposed digits. For example, the 8 above comes from 1 - 9 and the 4 from 5 - 1.
The result 10948 starts with 1 as this 1 comes from 1 - 0 (the 0 being "invisible" though).
The box algebra is fun to explore – note for instance that one always has (a ■ b) = (b ■ a), and (a ■ b) ■ c = a ■ (b ■ c), etc. [NO for the associativity! Read Maximilian below!]
Here is a sequence S of numbers such as (n ■ k) is always a square, k being the smallest possible integer:
S = 2, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 5, 6, 7, 8, 9, 21, 1, 2, 3, 4, 6, 7, 8, 9, 19, 10, 11, 1, 2, 3, 9, 17, 18, 19, 29, 20, 10, 11, 12, 13, 19, 27, 28, 29, 39, 30, 20, 21, 22, 10, 4, 5, 6, 7, 8, 1, ...
Example:
For n = 1 the smallest k producing a square is 2 (as 1 ■ 2 = 1, this 1 being the square of 1);
For n = 2 the smallest k producing a square is 1 (as 2 ■ 1 = 1, this 1 being the square of 1);
For n = 3 the smallest k producing a square is 2 (as 3 ■ 2 = 1, this 1 being the square of 1);
For n = 4 the smallest k producing a square is 3 (as 4 ■ 3 = 1, this 1 being the square of 1);
For n = 5 the smallest k producing a square is 3 (as 5 ■ 1 = 4, this 4 being the square of 2);
For n = 6 the smallest k producing a square is 3 (as 6 ■ 2 = 4, this 4 being the square of 2);
...
For n = 16 the smallest k producing a square is 12 (as 16 ■ 12 = 4, this 4 being the square of 2);
For n = 17 the smallest k producing a square is 1 (as 17 ■ 1 = 16, this 16 being the square of 4);
etc.
My friend Jean-Marc Falcoz has computed the first 20000 terms of S. He writes me that the highest k is 2175 so far (with 16575 ■ 2175 = 14400, square of 120) – the missing ks so far being 565, 678, 680, 681, etc.
The graph of S is emblematic of the rainfalls that affect Brussels for at least another week!
Have fun until the ■ Boxing Day!
Best,
É.
Very funny! Yes, the ■ operation has the nice properties of being commutative, associative, and having a neutral element e = 0 (x ■ 0 = 0 ■ x = x) and every element x has a symmetric x' such that x ■ x' = e, namely itself: x' = x. This makes it an abelian group. So it is natural to think of it as a fancy law of addition. One might want to find a "compatible" multiplication, i.e., distributive over ■. Then it might be natural to consider squares for that multiplication, rather than the usual one.
RépondreSupprimer(The property x+x=0 holds in [https://en.wikipedia.org/wiki/Boolean_ring Boolean rings] characterized by x² = x for all x. But in that case, "squares" are obviously not very interesting...)
Oops, actually ■ is *not* associative as you wrote [(a ■ b) ■ c =?= a ■ (b ■ c)], consider e.g., a=2, b=c=1. [Else you could find the a(n)=k you are looking for as min { x² ■ n, y² ■ n} \ {0}, where x² is the largest square < n and y² is the smallest square > n, I think.]
SupprimerGeeee, glad there is someone reading my posts! Thank you Maximilian!
Supprimer