A261176 Minimum value of (1/2)*Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) * ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

0, 9, 126, 802, 3158, 10040, 25464, 58837, 123422, 238203, 429467, 733923, 1200319, 1912928, 2945116, 4369570, 6338678, 9053512, 12622814, 17359779, 23503546, 31347788, 41161317

Offset

1,2

Comments

In one of his programming contests, Al Zimmermann coined the term "Delacorte Numbers" (after G. T. Delacorte, Jr., a New York City philantropist and benefactor) for the sum of D(a,b) = gcd(a,b) * distance^2(a,b), taken over all distinct pairs of integers (a,b) in a rectangular matrix.

The challenge in the contest was to find two kinds of arrangements of 1 to n^2, one minimizing the combined sum (this sequence) and the other maximizing the combined sum (A261177).

All terms beyond a(5) are conjectured based on numerical results. Terms up to a(17) have at least 5 independent verifications.

Upper bounds for the next terms are a(24)<=53670478, a(25)<=68938808, a(26)<=87777189, a(27)<=110759499.

Links

Table of n, a(n) for n=1..23.

Al Zimmermann's Programming Contests, Delacorte Numbers, Description, October 2014.

Al Zimmermann's Programming Contests, Delacorte Numbers, Final Report, January 2015.

The New York Community Trust: George T. Delacorte.

Arch D. Robison, Computing Delacorte Numbers with Julia, January 21, 2015.

Example

a(2)=9, because the matrix ((1 2)(3 4)) has Delacorte Number
D(1,2) + D(1,3) + D(1,4) + D(2,3) + D(2,4) + D(3,4) =
gcd(1,2)*(1^2 + 0^2) +
gcd(1,3)*(0^2 + 1^2) +
gcd(1,4)*(1^2 + 1^2) +
gcd(2,3)*(1^2 + 1^2) +
gcd(2,4)*(0^2 + 1^2) +
gcd(3,4)*(1^2 + 0^2) = 1*1 + 1*1 + 1*2 + 1*2 + 2*1 + 1*1 = 9.
Putting (2,4) in a row or column gives the minimum value of the matrix, whereas putting this pair in one of the diagonals gives the maximum.
a(3)=126, because no arrangement of the matrix elements exists that produces a smaller Delacorte Number than e.g. ((1 2 4)(3 6 8)(5 9 7)).

Crossrefs

Cf. A261177, A003989, A018782.

Sequence in context: A224495 A064199 A092343 * A261743 A229283 A144073

Adjacent sequences: A261173 A261174 A261175 * A261177 A261178 A261179

Keyword

nonn,hard

Author

Hugo Pfoertner, Aug 15 2015

Status

approved