|
%I #23 Oct 29 2021 12:26:46
%S 0,2,5,10,16,23,31,40,48,58,67,79,89,99
%N 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.
%C This sequence uses the same rules as A346069 except that here integer division instead of multiplication is used. See that sequence for further details.
%C The maximum sum of the quotients generally occurs when the smaller and larger numbers lie on two offset diagonal grids. See the examples below.
%e 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.
%e 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:
%e .
%e 4 1
%e 2 3
%e .
%e The quotient sum is then (4/1) + (4/2) + (3/1) + (3/2) = 10.
%e a(13) = 89. One way to arrange the numbers is:
%e .
%e 7
%e 6 12 2 8
%e 11 1 13 4
%e 5 10 3 9
%e .
%e 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.
%Y Cf. A346069 (multiplication), A348090 (addition), A003991, A003056.
%K nonn,more
%O 1,2
%A _Scott R. Shannon_, Oct 23 2021
| 