

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



0, 2, 5, 10, 16, 23, 31, 40, 48, 58, 67, 79, 89, 99
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


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.


LINKS



EXAMPLE

a(3) = 5. The numbers 1,2,3 can be placed next to each other in six ways: 123, 132, 213, 231, 312, 321. The combinations with the largest pair quotient sums are 312 and 213, 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



KEYWORD

nonn,more


AUTHOR



STATUS

approved



