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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A384222 Irregular triangle read by rows: T(n,k) is the length of the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 6, 1, 1, 2, 2, 1, 2, 1, 5, 1, 1, 6, 1, 1, 6, 1, 1, 1, 1, 2, 2, 1, 1, 8, 1, 1, 1, 2, 2, 1, 1, 1, 1, 6, 1, 1, 8, 1, 1, 6, 1, 1, 1, 1, 2, 2, 1, 2, 1, 9, 1, 1, 2, 2, 1, 1, 1, 1, 8, 1, 1, 8, 1, 1, 3, 3, 1, 4, 1, 2, 2, 1, 1, 10, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 8
Offset: 1

Views

Author

Omar E. Pol, Jun 03 2025

Keywords

Comments

The 2-dense sublists of divisors of n are the maximal sublists 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.
Row n has only one term, which is A000005(n), if and only if n is in A174973.
Conjecture 1: row n is a palindromic composition of A000005(n).
If the conjecture is true then this triangle should be a companion of A237270 in the sense that here the n-th row should be a palindromic composition of sigma_0(n) = A000005(n) and the n-th row of A237270 is a palindromic composition of sigma_1(n) = A000203(n).
A384149(n,k) is the sum of the terms in the k-th sublist of divisors of n. In the comments of A384149 it is conjectured that the row lengths of that triangle give A237271. If that conjecture is true then here the row lengths should also be A237271 and therefore A237271(n) could be defined also as the number of 2-dense sublists of divisors of n.

Examples

			  ----------------------------------------------------------------
  |  n | Row n of     |  List of divisors of n       | Number of |
  |    | the triangle |  [with sublists in brackets] | sublists  |
  ----------------------------------------------------------------
  |  1 |  1;          |  [1];                        |     1     |
  |  2 |  2;          |  [1, 2];                     |     1     |
  |  3 |  1, 1;       |  [1], [3];                   |     2     |
  |  4 |  3;          |  [1, 2, 4];                  |     1     |
  |  5 |  1, 1;       |  [1], [5];                   |     2     |
  |  6 |  4;          |  [1, 2, 3, 6];               |     1     |
  |  7 |  1, 1;       |  [1], [7];                   |     2     |
  |  8 |  4;          |  [1, 2, 4, 8];               |     1     |
  |  9 |  1, 1, 1;    |  [1], [3], [9];              |     3     |
  | 10 |  2, 2;       |  [1, 2], [5, 10];            |     2     |
  | 11 |  1, 1;       |  [1], [11];                  |     2     |
  | 12 |  6;          |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |  1, 1;       |  [1], [13];                  |     2     |
  | 14 |  2, 2;       |  [1, 2], [7, 14];            |     2     |
  | 15 |  1, 2, 1;    |  [1], [3, 5], [15];          |     3     |
  | 16 |  5;          |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |  1, 1;       |  [1], [17];                  |     2     |
  | 18 |  6;          |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |  1, 1;       |  [1], [19];                  |     2     |
  | 20 |  6;          |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |  1, 1, 1, 1; |  [1], [3], [7], [21];        |     4     |
  | 22 |  2, 2;       |  [1, 2], [11, 22];           |     2     |
  | 23 |  1, 1;       |  [1], [23];                  |     2     |
  | 24 |  8;          |  [1, 2, 3, 4, 6, 8, 12, 24]; |     1     |
   ...
  ...
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 two terms, so the row 14 is [2, 2].
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 terms in the sublists are [1, 2, 1] respectively, so the row 15 is [1, 2, 1].
78 is the first practical number A005153 not in A174973. For n = 78 the list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two sublists of divisors of 78 whose terms increase by a factor of at most 2, they are [1, 2, 3, 6] and [13, 26, 39, 78]. The number of terms in the sublists are [4, 4] respectively, so the row 78 is [4, 4].
From _Omar E. Pol_, Jul 23 2025: (Start)
A visualization with symmetries of the list of divisors of the first 24 positive integers and the sublists of divisors is as shown below:
  ---------------------------------------------------------------------------------
  |     n     |                 List of divisors of n                 | Number of |
  |           |        [with sublists of divisors in brackets]        | sublists  |
  ---------------------------------------------------------------------------------
  |     1     |                          [1]                          |     1     |
  |     2     |                         [1 2]                         |     1     |
  |     3     |                        [1] [3]                        |     2     |
  |     4     |                       [1  2  4]                       |     1     |
  |     5     |                      [1]     [5]                      |     2     |
  |     6     |                     [1   2 3   6]                     |     1     |
  |     7     |                    [1]         [7]                    |     2     |
  |     8     |                   [1    2   4    8]                   |     1     |
  |     9     |                  [1]     [3]     [9]                  |     3     |
  |    10     |                 [1     2]   [5    10]                 |     2     |
  |    11     |                [1]                [11]                |     2     |
  |    12     |               [1      2  3 4  6     12]               |     1     |
  |    13     |              [1]                    [13]              |     2     |
  |    14     |             [1       2]       [7      14]             |     2     |
  |    15     |            [1]         [3   5]        [15]            |     3     |
  |    16     |           [1        2     4     8       16]           |     1     |
  |    17     |          [1]                            [17]          |     2     |
  |    18     |         [1         2   3     6   9        18]         |     1     |
  |    19     |        [1]                                [19]        |     2     |
  |    20     |       [1          2      4 5     10         20]       |     1     |
  |    21     |      [1]             [3]     [7]            [21]      |     4     |
  |    22     |     [1           2]              [11          22]     |     2     |
  |    23     |    [1]                                        [23]    |     2     |
  |    24     |   [1            2    3  4   6  8   12           24]   |     1     |
       ...
A similar structure show the positive integers in the square array A385000. (End)

Links

Crossrefs

Row sums give A000005.
Cf. A000203, A005153, A027750, A174973 (2-dense numbers), A237271, A379288, A384149, A384225, A384226, A384928, A384930, A384931, A385000, A386984, A386989, A387030.

Programs

  • Mathematica
    A384222row[n_] := Map[Length, Split[Divisors[n], #2/# <= 2 &]];
Array[A384222row, 50] (* Paolo Xausa, Jul 08 2025 *)

A384225 Irregular triangle read by rows: T(n,k) is the number of odd divisors in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jun 16 2025

Keywords

Comments

T(n,k) is the number of odd numbers 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.

Examples

			  ------------------------------------------------------------------
  |  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].

Links

Crossrefs

Row sums give A001227.
Cf. A000203, A027750, A174973 (2-dense numbers), A280940, A237271, A379288, A384149, A384222, A384226, A384928, A384930, A384931, A385000, A386984, A386989, A387030.

Programs

  • Mathematica
    A384225row[n_] := Map[Count[#, _?OddQ] &, Split[Divisors[n], #2/# <= 2 &]];
Array[A384225row, 50] (* Paolo Xausa, Jul 08 2025 *)

A386992 Irregular triangle read by rows: T(n,k) is the number of nonprimes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 3, 1, 0, 1, 1, 1, 1, 0, 4, 1, 0, 1, 1, 1, 0, 1, 4, 1, 0, 4, 1, 0, 4, 1, 0, 0, 1, 1, 1, 1, 0, 6, 1, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 0, 5, 1, 0, 5, 1, 0, 0, 1, 1, 1, 1, 0, 1, 7, 1, 0, 1, 1, 1, 0, 0, 1, 6, 1, 0, 5, 1, 0, 2, 2, 1, 2, 1, 1, 1, 1, 0, 8, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0
Offset: 1

Views

Author

Omar E. Pol, Aug 23 2025

Keywords

Comments

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.
The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
It is conjectured that row lengths are given by A237271.

Examples

			Triangle begins:
  1;
  1;
  1, 0;
  2;
  1, 0;
  2;
  1, 0;
  3;
  1, 0, 1;
  1, 1;
  1, 0;
  4;
  1, 0;
  1, 1;
  1, 0, 1;
  ...
For n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2] and [5, 10]. There is a nonprime number in each sublist, so row 10 is [1, 1].
For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15]. Only the first and the third sublists contain nonprimes, so row 15 is [1, 0, 1].

Links

Crossrefs

Row sums give A033273.
Cf. A018252, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386993, A387030.

Programs

  • Mathematica
    A386992row[n_] := Map[Count[#, _?(!PrimeQ[#] &)] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A386992row, 50] (* Paolo Xausa, Aug 28 2025 *)

Formula

T(n,k) = A384222(n,k) - A387030(n,k).

A386993 Number of 2-dense sublists of divisors of the n-th squarefree number.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 1, 2, 4, 2, 3, 2, 2, 4, 2, 1, 2, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 4, 1, 2, 4, 3, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 4, 2, 3, 4, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 3, 4, 2, 2, 4, 2, 4, 2, 4, 2, 4, 3, 2, 4, 2, 2, 2, 2, 4, 2, 3, 4, 2, 2, 2, 3, 4, 2, 2, 4, 4, 2, 5, 2, 2, 3, 2
Offset: 1

Views

Author

Omar E. Pol, Aug 23 2025

Keywords

Comments

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

Examples

			For n = 11 the 11th squarefree number is 15. The list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15], so a(11) = 3.

Links

Crossrefs

Cf. A005117, A174973 (2-dense numbers), A237271, A379288, A380328, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386992, A387030.

Programs

  • Mathematica
    Map[Length[Split[Divisors[#], #2 <= 2*# &]] &, Select[Range[150], SquareFreeQ]] (* Paolo Xausa, Aug 29 2025 *)

Formula

a(n) = A237271(A005117(n)). (conjectured).

A386994 Number of 2-dense sublists of divisors of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 4, 2, 4, 2, 1, 2, 4, 4, 8, 2, 3, 4, 8, 4, 4, 2, 1, 6, 4, 4, 12, 2, 1, 4, 16, 4, 4, 8, 1, 8, 8, 4, 3, 4, 1, 2, 11, 6, 8, 2, 1, 8, 10, 4, 12, 4, 3, 13, 5, 10, 8, 4, 1, 4, 8, 10, 17, 8, 7, 8, 20, 9, 15, 4, 1, 4, 16, 18, 24, 15, 7, 4, 3, 5
Offset: 0

Views

Author

Omar E. Pol, Aug 27 2025

Keywords

Comments

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

Examples

			For n = 18 the 18th Fibonacci number is 2584. The list of divisors of 2584 is [1, 2, 4, 8, 17, 19, 34, 38, 68, 76, 136, 152, 323, 646, 1292, 2584]. There are three 2-dense sublists of divisors of 2584, they are [1, 2, 4, 8], [17, 19, 34, 38, 68, 76, 136, 152] and [323, 646, 1292, 2584], so a(18) = 3.

Crossrefs

Cf. A000045, A133021, A139045, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386992, A387030, A386993.

Programs

  • Mathematica
    A386994[n_] := Length[Split[Divisors[Fibonacci[n]], #2 <= 2*# &]];
Array[A386994, 100, 0] (* Paolo Xausa, Sep 02 2025 *)

Formula

a(n) = A237271(A000045(n)), n >= 1. (conjectured).

Extensions

More terms from Alois P. Heinz, Aug 27 2025
Showing 1-5 of 5 results.

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