[seqfan] Re: Boxing days

[Neil Sloane]

Eric,
I will create an entry - it will be A329794
Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sat, Dec 7, 2019 at 11:55 AM Éric Angelini <bk263401 at skynet.be> wrote:

> 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.
>
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