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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.

%I A348387 #23 Oct 29 2021 12:26:46
%S A348387 0,2,5,10,16,23,31,40,48,58,67,79,89,99
%N 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.
%C A348387 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 A348387 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 A348387 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 A348387 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 A348387 .
%e A348387   4 1
%e A348387   2 3
%e A348387 .
%e A348387 The quotient sum is then (4/1) + (4/2) + (3/1) + (3/2) = 10.
%e A348387 a(13) = 89. One way to arrange the numbers is:
%e A348387 .
%e A348387            7
%e A348387    6  12   2   8
%e A348387   11   1  13   4
%e A348387    5  10   3   9
%e A348387 .
%e A348387 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 A348387 Cf. A346069 (multiplication), A348090 (addition), A003991, A003056.
%K A348387 nonn,more
%O A348387 1,2
%A A348387 _Scott R. Shannon_, Oct 23 2021