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

0,3

Comments

These are row sums of A332080, with terms interpreted as base n+1 numbers.

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.

a(6) <= 331 and a(7) <= 747, cf. examples in A332080.

Links

Table of n, a(n) for n=0..5.

Example

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.

Prog

(PARI) A332081(n)=if(n, vecsum([fromdigits(digits(x), n+1)|x<-concat(A332080(n))]))

Crossrefs

Cf. A332080.

Sequence in context: A273314 A281424 A005745 * A213557 A273386 A045618

Adjacent sequences: A332078 A332079 A332080 * A332082 A332083 A332084

Keyword

nonn,hard,more,base

Author

M. F. Hasler, Aug 17 2020

Status

approved