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.

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, 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, 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

Offset

1,6

Links

Table of n, a(n) for n=1..95.

Example

For n = 1..17 every row of the triangle has only one term.
For n = 18..30 the triangle is as shown below:
  3, 9;
  0;
  5;
  0;
  0;
  0;
  3;
  0;
  0;
  0;
  7;
  0;
  3, 5, 15;
  ...
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:
   d  <  m  <  2*d
--------------------
   1            2
   2     3      4
   3     5      6
   5           10
   6           12
  10    15     20
  15           30
  30           60
.
So the 30th row of the triangle is [3, 5, 15].
.
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:
   d  <  m  <  2*d
--------------------
   1            2
   2     3      4
   3            6
   6           12
  13           26
  26    39     52
  39           78
  78          156
.
Note that 13 is an odd divisor of 78 but 13 does not qualify.
So the 78th row of the triangle is [3, 39].

Mathematica

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 *)

Crossrefs

Also zeros and odd terms of A379461.

Row sums give A383147.

The number of positive terms in row n is A239657(n).

Cf. A027750, A237593, A379288, A379374.

Sequence in context: A360016 A035678 A094518 * A383147 A123074 A095269

Adjacent sequences: A383206 A383207 A383208 * A383210 A383211 A383212

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 19 2025

Status

approved