cp's OEIS Frontend

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.

%I A332081 #16 Sep 17 2020 12:03:55
%S A332081 0,1,6,23,67,159
%N 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.
%C A332081 These are row sums of A332080, with terms interpreted as base n+1 numbers.
%C A332081 Rows of A332080 are indeed the nonzero diagonals, read by increasing length, of such triangular matrices. In A332080 the nonzero part of these matrices is seen as a triangle; i.e., rotated by 45 degrees, say. None of the elements of this part may be zero, but they may have (not leading) digits zero. The elements of the vanishing part of the matrix are ignored or considered to have no digit.
%C A332081 a(6) <= 331 and a(7) <= 747, cf. examples in A332080.
%e A332081 For n = 0 the matrix and sum of its elements are empty, so a(0) = 0.
%e A332081 For n = 1 the matrix is [1], so a(1) = 1.
%e A332081 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).
%e A332081 For n = 5, one such triangular matrix with minimal sum is given as follows:
%e A332081     1   2   3   4   5
%e A332081     .  15  20  33  44
%e A332081     .   .  14  25  30
%e A332081     .   .   .  10  22
%e A332081     .   .   .   .  11
%e A332081 where all numbers are written in base 6.
%e A332081 One easily checks that no two entries in any row or column have a digit in common.
%e A332081 The sum of these base-6 numbers (e.g., 44[6] = 4*6 + 4 = 28) is a(5) = 159.
%e A332081 There is no such triangle with a smaller sum.
%o A332081 (PARI) A332081(n)=if(n,vecsum([fromdigits(digits(x),n+1)|x<-concat(A332080(n))]))
%Y A332081 Cf. A332080.
%K A332081 nonn,hard,more,base
%O A332081 0,3
%A A332081 _M. F. Hasler_, Aug 17 2020