%I #19 Mar 29 2019 13:07:51

%S 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,

%T 8,4,9

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

%C "Pairwise products" includes the squares of the digits.

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

%C 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).

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

%C 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.

%e 1012014015216242530327281835456364849.

%Y A203565 considers only products of adjacent digits.

%K nonn,cons,fini,full,base

%O 37,4

%A _Rob Pratt_, Mar 27 2019