A348387 Place the numbers 1 to n on a square grid and for all created orthogonally adjacent pairs divide the larger value by the smaller, using integer division; a(n) gives the maximum possible value of the sum of all pair quotients.

0, 2, 5, 10, 16, 23, 31, 40, 48, 58, 67, 79, 89, 99

Offset: 1

Author

Scott R. Shannon, Oct 23 2021

Keywords

• nonn
• more

Comments

This sequence uses the same rules as A346069 except that here integer division instead of multiplication is used. See that sequence for further details.

The maximum sum of the quotients generally occurs when the smaller and larger numbers lie on two offset diagonal grids. See the examples below.

Examples

			a(3) = 5. The numbers 1,2,3 can be placed next to each other in six ways: 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, 3-2-1. The combinations with the largest pair quotient sums are 3-1-2 and 2-1-3, the sum being (3/1) + (2/1) = 5.
a(4) = 10. The best way to arrange the numbers is in a 2 X 2 square where 4 is on opposite corner to the 3:
.
  4 1
  2 3
.
The quotient sum is then (4/1) + (4/2) + (3/1) + (3/2) = 10.
a(13) = 89. One way to arrange the numbers is:
.
           7
   6  12   2   8
  11   1  13   4
   5  10   3   9
.
The quotient sum is then (12/6) + (12/2) + (8/2) + (11/1) + (13/1) + (13/4) +(10/5) + (10/3) + (9/3) + (11/6) + (11/5) + (12/1) + (10/1) + (7/2) + (13/2) + (13/3) + (8/4) + (9/4) = 89. Note how the smaller and larger numbers lie on offset diagonal grids.
		

Crossrefs

Cf. A346069 (multiplication), A348090 (addition), A003991, A003056.