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A384225
Irregular triangle read by rows: T(n,k) is the number of odd numbers in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.
28
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4
OFFSET
1,8
COMMENTS
T(n,k) is the number of odd divisors in the k-th sublist of divisors of n whose terms increase by a factor of at most 2,
In a sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
At least for the first 1000 rows the row lengths give A237271.
Observation: at least the first 33 rows (or first 62 terms) coincide with A280940.
Is this the same as A280940 ? - Omar E. Pol, Nov 05 2025
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).
FORMULA
T(n,k) = A384222(n,k) - A386604(n,k). - Omar E. Pol, Nov 05 2025
EXAMPLE
------------------------------------------------------------------
| n | Row n of | List of divisors of n | Number of |
| | the triangle | [with sublists in brackets] | sublists |
------------------------------------------------------------------
| 1 | 1; | [1]; | 1 |
| 2 | 1; | [1, 2]; | 1 |
| 3 | 1, 1; | [1], [3]; | 2 |
| 4 | 1; | [1, 2, 4]; | 1 |
| 5 | 1, 1; | [1], [5]; | 2 |
| 6 | 2; | [1, 2, 3, 6]; | 1 |
| 7 | 1, 1; | [1], [7]; | 2 |
| 8 | 1; | [1, 2, 4, 8]; | 1 |
| 9 | 1, 1, 1; | [1], [3], [9]; | 3 |
| 10 | 1, 1; | [1, 2], [5, 10]; | 2 |
| 11 | 1, 1; | [1], [11]; | 2 |
| 12 | 2; | [1, 2, 3, 4, 6, 12]; | 1 |
| 13 | 1, 1; | [1], [13]; | 2 |
| 14 | 1, 1; | [1, 2], [7, 14]; | 2 |
| 15 | 1, 2, 1; | [1], [3, 5], [15]; | 3 |
| 16 | 1; | [1, 2, 4, 8, 16]; | 1 |
| 17 | 1, 1; | [1], [17]; | 2 |
| 18 | 3; | [1, 2, 3, 6, 9, 18]; | 1 |
| 19 | 1, 1; | [1], [19]; | 2 |
| 20 | 2; | [1, 2, 4, 5, 10, 20]; | 1 |
| 21 | 1, 1, 1, 1; | [1], [3], [7], [21]; | 4 |
...
For n = 14 the list of divisors of 14 is [1, 2, 7, 14]. There are two sublists of divisors of 14 whose terms increase by a factor of at most 2, they are [1, 2] and [7, 14]. Each sublist has only one odd number, so the row 14 is [1, 1].
For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three sublists of divisors of 15 whose terms increase by a factor of at most 2, they are [1], [3, 5], [15]. The number of odd numbers in the sublists are [1, 2, 1] respectively, so the row 15 is [1, 2, 1].
For n = 16 the list of divisors of 16 is [1, 2, 4, 8, 16]. There is only one sublist of divisors of 16 whose terms increase by a factor of at most 2, that is the same as the list of divisors of 16, which has five terms and only one odd number, so the row 16 is [1].
MATHEMATICA
A384225row[n_] := Map[Count[#, _?OddQ] &, Split[Divisors[n], #2/# <= 2 &]];
Array[A384225row, 50] (* Paolo Xausa, Jul 08 2025 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Jun 16 2025
STATUS
approved