[seqfan] Boxing days

[Éric Angelini]

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).

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!
Best,
É.