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A383209 Irregular triangle read by rows in which row n lists the odd divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such odd divisors do not exist.

%I A383209 #26 Apr 27 2025 15:06:20
%S A383209 0,0,0,0,0,3,0,0,0,0,0,3,0,0,5,0,0,3,9,0,5,0,0,0,3,0,0,0,7,0,3,5,15,0,
%T A383209 0,0,0,7,3,9,0,0,0,5,0,3,7,21,0,0,5,9,15,0,0,3,0,0,0,0,0,3,9,27,0,7,0,
%U A383209 0,0,3,5,15,0,0,9,0,0,3,11,33,0,0,0,7,0,3,9,0,0,5,25,0,11,3,39
%N A383209 Irregular triangle read by rows in which row n lists the odd divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such odd divisors do not exist.
%e A383209 For n = 1..17 every row of the triangle has only one term.
%e A383209 For n = 18..30 the triangle is as shown below:
%e A383209   3, 9;
%e A383209   0;
%e A383209   5;
%e A383209   0;
%e A383209   0;
%e A383209   0;
%e A383209   3;
%e A383209   0;
%e A383209   0;
%e A383209   0;
%e A383209   7;
%e A383209   0;
%e A383209   3, 5, 15;
%e A383209   ...
%e A383209 For n = 30 there are three odd divisors m of 30 such that there is a divisor d of 30 with d < m < 2*d. Those odd divisors are 3, 5 and 15 as shown below:
%e A383209    d  <  m  <  2*d
%e A383209 --------------------
%e A383209    1            2
%e A383209    2     3      4
%e A383209    3     5      6
%e A383209    5           10
%e A383209    6           12
%e A383209   10    15     20
%e A383209   15           30
%e A383209   30           60
%e A383209 .
%e A383209 So the 30th row of the triangle is [3, 5, 15].
%e A383209 .
%e A383209 For n = 78 there are two odd divisors m of 78 such that there is a divisor d of 78 with d < m < 2*d. Those odd divisors are 3 and 39 as shown below:
%e A383209    d  <  m  <  2*d
%e A383209 --------------------
%e A383209    1            2
%e A383209    2     3      4
%e A383209    3            6
%e A383209    6           12
%e A383209   13           26
%e A383209   26    39     52
%e A383209   39           78
%e A383209   78          156
%e A383209 .
%e A383209 Note that 13 is an odd divisor of 78 but 13 does not qualify.
%e A383209 So the 78th row of the triangle is [3, 39].
%t A383209 row[n_] := Module[{d = Partition[Divisors[n], 2, 1], r}, r = Select[d, OddQ[#[[2]]] && #[[2]] < 2*#[[1]] &][[;; , 2]]; If[r == {}, {0}, r]]; Array[row, 80] // Flatten (* _Amiram Eldar_, Apr 19 2025 *)
%Y A383209 Also zeros and odd terms of A379461.
%Y A383209 Row sums give A383147.
%Y A383209 The number of positive terms in row n is A239657(n).
%Y A383209 Cf. A027750, A237593, A379288, A379374.
%K A383209 nonn,tabf
%O A383209 1,6
%A A383209 _Omar E. Pol_, Apr 19 2025