### Smallest multiplication

Hello Math-Fun
We’re looking today for the “smallest” multiplication (using new distinct terms) that produces one after the other all the nonnegative integers.

0 is present in 0 x 1 = 0
1 is present in 2 x 5 = 10 (as we cannot use again the integers 0 and 1)
2 is present in 3 x 4 = 12 (as we cannot use again the integers 0, 1, 2 and 5)
3 is present in 6 x 22 = 132 (as we cannot use again the integers 0, 1, 2, 3, 4 and 5)
4 is present in 7 x 12 = 84 (as we cannot …)
5 is present in 8 x 19 = 152
6 is present in 9 x 14 = 126
7 is present in 10 x 17 = 170
8 is present in 11 x 18 = 198
9 is present in 13 x 15 = 195
10 is present in 16 x 63 = 1008
11 is present in 19 x 58 = 1102

Integers used so far that cannot be reused (if I’m not wrong):
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 58, 63

We could submit S to the OEIS, with the hereunder (heavy) definition:

Definition
— The sequence S must be seen as a succession of pairs of integers:
[a(1), a(2)], [a(3),a(4)], [a(5), a(6)], [a(7), a(8)],…
— All a(n)s are distinct nonnegative integers.
— The integer resulting in a(1) * a(2) has “0” as substring; the product a(3) * a(4) has “1” as substring; the product a(5) * a(6) has “2” as substring; etc.

We have so far for S:

S = 0, 1, 2, 5, 3, 4, 6, 22, 7, 12, 8, 19, 9, 14, 10, 17, 11, 18, 13, 15, 16, 63, 19, 58, …
Best,
É.
__________
Update

Maximilian H. (the best typo/bug-hunter in town!) was quick to correct and extend S:

Hello Eric,

I confirm your terms upto (16, 63), but then I get (20, 55),
the 19 is used earlier in 8 * 19 :

S = (0, 1), (2, 5), (3, 4), (6, 22), (7, 12), (8, 19), (9, 14), (10, 17), (11, 18), (13, 15), (16, 63), (20, 55), (21, 58), (23, 31), (24, 59), (25, 46), (26, 62), (27, 64), (28, 65), (29, 66), (30, 34), (32, 38), (33, 37), (35, 67), (36, 68), (39, 75), (40, 315), (41, 47), (42, 69), (43, 103), (44, 70), (45, 71), (48, 84), (49, 117), (50, 268), (51, 85), (52, 93),  ...

> For the definition one could use :
Lexicographic earliest sequence of nonnegative integers without duplicates
such that S(2k) * S(2k+1) has k as substring, for all k = 0, 1, 2, ...

-M.
__________
Great (as always!) -- merci Maximilian!

### Commentaires

1. I get: S = (0, 1), (2, 5), (3, 4), (6, 22), (7, 12), (8, 19), (9, 14), (10, 17), (11, 18),
(13, 15), (16, 63), (20, 55), (21, 58), (23, 31), (24, 59), (25, 46), (26, 62),
(27, 64), (28, 65), (29, 66), (30, 34), (32, 38), (33, 37), (35, 67), (36, 68),
(39, 75), (40, 315), (41, 47), (42, 69), (43, 103), (44, 70), (45, 71), (48, 84),
(49, 117), (50, 268), (51, 85), (52, 93), ...
Alternate definition: Lexicographic earliest sequence of distinct nonnegative integers such that S(2k) * S(2k+1) has k as substring, for all k >= 0.