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A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

%I A299761 #43 Aug 04 2022 05:04:38
%S A299761 1,1,0,2,0,2,3,0,2,3,0,0,3,4,0,0,3,5,4,0,3,0,4,5,0,0,0,4,6,5,0,0,4,7,
%T A299761 0,5,6,0,4,0,0,5,7,6,0,0,0,5,8,0,6,7,0,0,5,9,0,0,6,8,7,5,0,0,0,6,9,0,
%U A299761 7,8,0,0,0,6,10,0,0,7,9,8,0,6,11,0,0,0,7,10,0,6,8,9,0,0,0,0,7,11,0,0,8,10
%N A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.
%C A299761 The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).
%H A299761 Michael De Vlieger, <a href="/A299761/b299761.txt">Table of n, a(n) for n = 1..14002</a> (rows 1 <= n <= 10^4)
%e A299761 Triangle begins (rows 1..16):
%e A299761 1;
%e A299761 1;
%e A299761 0;
%e A299761 2;
%e A299761 0;
%e A299761 2, 3;
%e A299761 0;
%e A299761 2;
%e A299761 3;
%e A299761 0;
%e A299761 0;
%e A299761 3, 4;
%e A299761 0;
%e A299761 0;
%e A299761 3, 5;
%e A299761 4;
%e A299761 ...
%e A299761 For n = 6 the middle divisors of 6 are 2 and 3, so row 6 is [2, 3].
%e A299761 For n = 7 there are no middle divisors of 7, so row 7 is [0].
%e A299761 For n = 8 the middle divisor of 8 is 2, so row 8 is [2].
%e A299761 For n = 72 the middle divisors of 72 are 6, 8 and 9, so row 72 is [6, 8, 9].
%t A299761 Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> {0}, {n, 80}] // Flatten (* _Michael De Vlieger_, Jun 14 2018 *)
%o A299761 (PARI) row(n) = my(v=select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); if (#v, v, [0]); \\ _Michel Marcus_, Aug 04 2022
%Y A299761 Row sums give A071090.
%Y A299761 The number of nonzero terms in row n is A067742(n).
%Y A299761 Nonzero terms give A303297.
%Y A299761 Indices of the rows where there are zeros give A071561.
%Y A299761 Indices of the rows where there are nonzero terms give A071562.
%Y A299761 Cf. A027750, A281007, A299777.
%K A299761 nonn,tabf,look
%O A299761 1,4
%A A299761 _Omar E. Pol_, Jun 08 2018