A307177 Decimal expansion of smallest nontrivial base-10 number that contains all pairwise products of its digits as substrings.

1, 0, 1, 2, 0, 1, 4, 0, 1, 5, 2, 1, 6, 2, 4, 2, 5, 3, 0, 3, 2, 7, 2, 8, 1, 8, 3, 5, 4, 5, 6, 3, 6, 4, 8, 4, 9

Offset: 37

Author

Rob Pratt, Mar 27 2019

Keywords

• nonn
• cons
• fini
• full
• base

Comments

"Pairwise products" includes the squares of the digits.

Suggested by Ricardo Palomino, who mentioned the trivial numbers A007088.

If any digit other than 0 or 1 appears, then all ten digits appear, as can easily be checked for each digit. For example, if 2 appears then 2*2 = 4 appears, which implies that 2*4 = 8 appears and {1,6} (from 4*4 = 16) appear, which implies that 3 appears (from 4*8 = 32), which implies that 3*3 = 9 appears, which implies that {2,7} appear (from 3*9), which implies that {5,6} appear (from 7*8), which implies that 0 appears (from 2*5 = 10).

There are 37 distinct products (10 with one digit and 27 with two digits) of pairs of digits from {0,1,...,9}.

Rob Pratt solved an asymmetric traveling salesman problem (ATSP) on 38 nodes to find the minimum number of digits, which turns out to be 37, and then solved a sequence of integer linear programming problems (minimizing one digit at a time from left to right) to find the minimum such 37-digit number.

Examples

			1012014015216242530327281835456364849.
		

Crossrefs

A203565 considers only products of adjacent digits.