A288350 Lexically smallest string of n digits from 1...9, such that no formula using the single digits in the given order exists that evaluates to 0.

1, 12, 124, 1251, 12721, 169896, 8985898

Offset: 1

Author

Hugo Pfoertner, Jun 08 2017

Keywords

• nonn
• base
• fini
• full

Comments

The formula may use any combination of the four binary operators +, -, *, /, the unary minus and parentheses. All digits have to be used exactly once and in the given order. Concatenation of the digits is not allowed. 0/0 in the evaluation of expressions makes the result invalid.

The last term a(7)=8985898, which is the only string of 7 digits for which it is not possible to construct an expression with result 0, is conjectured to be the last sequence term, i.e. for all strings of 8 or more digits expressions with result 0 can be found.

The conjecture is almost trivially true. Since 8985898 is the only exception for strings of 7 digits, an 8 digit string containing 8985898 can only be of the form x8985898 or 8985898x. But since any of x898589 and 985898x /= 8985898 leads to a 7 digit string that can represent 0, one can always find a substring of length 7 representing 0 in a string of 8 or more digits. Therefore the sequence ends at a(7).

Examples

			a(3)=124: For all numbers starting with 11.. the formula can be chosen as (1-1)*..,
121: 1-2+1=0, 122: (1-2/2)=0, 123: 1+2-3=0, 124: 0 cannot by obtained by any valid formula.
		

Links

• IBM Research, Ponder This Challenge - June 2017, related problem for 7-digit numbers.

Crossrefs

Cf. A000108 (number of ways to insert the parentheses), A181898, A288351, A288353, A288354, A288355, A288356.