A332081 Minimum sum of elements of an n X n upper triangular matrix of distinct positive numbers such that no two entries in the same row or column share a digit when the numbers are written in base n+1.
0, 1, 6, 23, 67, 159
Offset: 0
Examples
For n = 0 the matrix and sum of its elements are empty, so a(0) = 0.
For n = 1 the matrix is [1], so a(1) = 1.
For n = 2, a matrix with the required property is given by [ 1, 2; 0, 10 ], where elements are written in base 3. Obviously there can't be a 2 X 2 matrix with this property with smaller sum of elements, so a(2) = 1 + 2 + 3 = 6 (where 3 = 10[3], i.e., 10 in base 3).
For n = 5, one such triangular matrix with minimal sum is given as follows:
1 2 3 4 5
. 15 20 33 44
. . 14 25 30
. . . 10 22
. . . . 11
where all numbers are written in base 6.
One easily checks that no two entries in any row or column have a digit in common.
The sum of these base-6 numbers (e.g., 44[6] = 4*6 + 4 = 28) is a(5) = 159.
There is no such triangle with a smaller sum.
Crossrefs
Cf. A332080.
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