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Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n lists the divisors of n but every middle divisor is replaced with zero.
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%I #16 Aug 06 2024 22:01:04

%S 0,0,2,1,3,1,0,4,1,5,1,0,0,6,1,7,1,0,4,8,1,0,9,1,2,5,10,1,11,1,2,0,0,

%T 6,12,1,13,1,2,7,14,1,0,0,15,1,2,0,8,16,1,17,1,2,0,6,9,18,1,19,1,2,0,

%U 0,10,20,1,3,7,21,1,2,11,22,1,23,1,2,3,0,0,8,12,24

%N Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n lists the divisors of n but every middle divisor is replaced with zero.

%C The nonzero terms in row n are the nonmiddle divisors of n.

%C The nonmiddle divisors of n are here the divisors of n that are not in the half-open interval [sqrt(n/2), sqrt(n*2)).

%e Triangle begins:

%e 0;

%e 0, 2;

%e 1, 3;

%e 1, 0, 4;

%e 1, 5;

%e 1, 0, 0, 6;

%e 1, 7;

%e 1, 0, 4, 8;

%e 1, 0, 9;

%e 1, 2, 5, 10;

%e 1, 11;

%e 1, 2, 0, 0, 6, 12;

%e ...

%e For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12] and the middle divisors are [3, 4], but here the middle divisors are replaced with zeros, so the 12th row of the triangle is [1, 2, 0, 0, 6, 12].

%t row[n_] := Divisors[n] /. {x_?(Sqrt[n/2] <= # < Sqrt[2*n] &) -> 0}; Table[row[n], {n, 1, 24}] // Flatten (* _Amiram Eldar_, Jul 29 2024 *)

%Y Row sums give A302433.

%Y Nonzero terms give A375038.

%Y Row lengths give A000005.

%Y The number of zeros in row n is A067742(n).

%Y The number of nonzero terms in row n is A067743(n).

%Y Cf. A000005, A027750, A067742, A299761, A303297.

%K nonn,tabf,easy

%O 1,3

%A _Omar E. Pol_, Jul 28 2024