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A133568 Triangle: row n is least sequence of n positive integers such that A131655(n) distinct rational numbers are generated only from them using only +, -, * and / and each number in the row no more than once in a given expression.

%I A133568 #13 Sep 02 2021 19:27:45
%S A133568 1,2,3,2,5,22,2,5,22,309,2,5,22,309,10431
%N A133568 Triangle: row n is least sequence of n positive integers such that A131655(n) distinct rational numbers are generated only from them using only +, -, * and / and each number in the row no more than once in a given expression.
%C A133568 As in A131655, these operations are all considered to be the usual binary operations only; in particular, -x is not a permitted expression.
%C A133568 The list of rational numbers generated by row 5 does not contain a 1 or 0 and therefore it still might be possible to construct row 6 by using the numbers from row 5. Futhermore, one might conjecture that all subsequent rows fully contain the previous one. - _Mikelis Emils Mikelsons_, Sep 01 2021
%H A133568 Mikelis Emils Mikelsons, <a href="/A133568/a133568.txt">The list of rational numbers generated by row 5</a>
%H A133568 Mikelis Emils Mikelsons, <a href="/A131655/a131655.txt">List of expressions for row 5</a>
%e A133568 Row 3: 2, 5, 22 generates these A131655(3)=89 distinct rational numbers:
%e A133568 -108, -100, -66, -39, -34, -25, -108/5, -20, -39/2, -19, -17, -15, -12,
%e A133568 -17/2, -22/3, -6, -54/11, -4, -3, -12/5, -39/22, -1/4, -3/22, -2/17, 1/55,
%e A133568 2/27, 1/11, 5/44, 2/17, 3/22, 5/24, 5/22, 1/4, 7/22, 2/5, 5/11,
%e A133568 39/22, 2, 11/5, 49/22, 12/5, 5/2, 3, 22/7, 4, 22/5, 24/5, 54/11, 5,
%e A133568 56/11, 6, 32/5, 7, 22/3, 17/2, 44/5, 10, 11, 12, 27/2, 15, 16, 17, 19,
%e A133568 39/2, 20, 108/5, 22, 112/5, 24, 49/2, 25, 27, 29, 32, 34, 39, 44, 49,
%e A133568 54, 55, 66, 100, 108, 110, 112, 120, 154, 220.
%e A133568 No smaller sequence of three positive integers generates this many distinct results.
%e A133568 Triangle begins:
%e A133568   1;
%e A133568   2, 3;
%e A133568   2, 5, 22;
%e A133568   2, 5, 22, 309;
%e A133568   2, 5, 22, 309, 10431;
%e A133568   ...
%Y A133568 Cf. A131655.
%K A133568 nonn,tabl,hard,more
%O A133568 1,2
%A A133568 _Rick L. Shepherd_, Sep 16 2007
%E A133568 More terms by _Mikelis Emils Mikelsons_, Sep 01 2021