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T(n, k) = floor(n/m) where m is the least positive integer such that floor(n/m) = floor(k/m). Square array read by antidiagonals, for n >= 0 and k >= 0.
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%I #15 Feb 04 2020 21:25:00

%S 0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,1,1,0,0,0,0,0,3,0,0,0,0,0,0,1,1,0,

%T 0,0,0,0,0,1,4,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,1,5,1,0,0,0,0,0,0,

%U 0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,6,1,0

%N T(n, k) = floor(n/m) where m is the least positive integer such that floor(n/m) = floor(k/m). Square array read by antidiagonals, for n >= 0 and k >= 0.

%C For any n > 0, the n-th row has A001651(n) nonzero terms.

%H Rémy Sigrist, <a href="/A331902/b331902.txt">Table of n, a(n) for n = 0..10010</a> (antidiagonals 0..140)

%H Rémy Sigrist, <a href="/A331886/a331886_1.png">Colored representation of T(n, k) for n, k = 0..1000</a> (where the hue is function of T(n, k), red pixels correspond to 0's)

%F T(n, k) = floor(n/A331886(n, k)) = floor(k/A331886(n, k)).

%F T(n, k) = T(k, n).

%F T(n, k) = 0 iff max(n, k) >= 2*min(n, k).

%F T(n, n+1) = A213633(n+1).

%e Array T(n, k) begins (with dots instead of 0's for readability):

%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12

%e ---+----------------------------------------------------

%e 0| . . . . . . . . . . . . .

%e 1| . 1 . . . . . . . . . . .

%e 2| . . 2 1 . . . . . . . . .

%e 3| . . 1 3 1 1 . . . . . . .

%e 4| . . . 1 4 2 1 1 . . . . .

%e 5| . . . 1 2 5 1 1 1 1 . . .

%e 6| . . . . 1 1 6 3 2 1 1 1 .

%e 7| . . . . 1 1 3 7 2 1 1 1 1

%e 8| . . . . . 1 2 2 8 4 2 2 1

%e 9| . . . . . 1 1 1 4 9 3 3 1

%e 10| . . . . . . 1 1 2 3 10 5 2

%e 11| . . . . . . 1 1 2 3 5 11 2

%e 12| . . . . . . . 1 1 1 2 2 12

%o (PARI) T(n,k) = for (x=1, oo, if (n\x==k\x, return (n\x)))

%Y Cf. A001651, A213633, A331886.

%K nonn,tabl

%O 0,13

%A _Rémy Sigrist_, Jan 31 2020