A379288 -id:A379288 - cp's OEIS frontend

A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

1, 2, 1, 3, 8, 1, 5, 36, 1, 7, 64, 1, 3, 9, 2, 50, 1, 11, 1728, 1, 13, 2, 98, 1, 15, 15, 1024, 1, 17, 5832, 1, 19, 8000, 1, 3, 7, 21, 2, 242, 1, 23, 331776, 1, 5, 25, 2, 338, 1, 3, 9, 27, 21952, 1, 29, 810000, 1, 31, 32768, 1, 3, 11, 33, 2, 578, 1, 35, 35, 10077696, 1, 37, 2, 722, 1, 3, 13, 39, 2560000

Offset: 1

Omar E. Pol, Aug 12 2025

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.

			Triangle begins:
   1;
   2;
   1,  3;
   8;
   1,  5;
  36;
   1,  7;
  64;
   1,  3,  9;
   2, 50;
  ...
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]. The product of terms are 1*2 = 2 and 5*10 = 50 respectively, so the row 10 of the triangle is [2, 50].

Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

Row products give A007955.

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

A386989row[n_] :=Times @@@ Split[Divisors[n], #2/# <= 2 &];
Array[A386989row, 50] (* Paolo Xausa, Aug 29 2025 *)

A379384 Sum of the divisors of n except the "e" divisors described in A005279.

Original entry on oeis.org

1, 3, 4, 7, 6, 9, 8, 15, 13, 18, 12, 15, 14, 24, 19, 31, 18, 27, 20, 37, 32, 36, 24, 27, 31, 42, 40, 49, 30, 33, 32, 63, 48, 54, 41, 39, 38, 60, 56, 67, 42, 65, 44, 84, 49, 72, 48, 51, 57, 93, 72, 98, 54, 81, 72, 91, 80, 90, 60, 63, 62, 96, 95, 127, 84, 97, 68

Offset: 1

Omar E. Pol, Dec 22 2024

nonn

Shares infinitely many terms with A000203.
a(n) = A000203(n) if n is not in A005279.
a(n) < A000203(n) if n is in A005279.

Row sums of A379374.

Cf. A000203, A005279, A237271, A237593, A379288, A379379.

a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, 1 + Total[Select[d, #[[2]] >= 2*#[[1]] &][[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 22 2024 *)

More terms from Alois P. Heinz, Dec 22 2024

A379461 Irregular triangle read by rows in which row n lists the divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such divisors do not exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 4, 6, 0, 0, 5, 0, 0, 3, 9, 0, 5, 0, 0, 0, 3, 4, 6, 8, 12, 0, 0, 0, 7, 0, 3, 5, 6, 10, 15, 0, 0, 0, 0, 7, 3, 4, 6, 9, 12, 18, 0, 0, 0, 5, 8, 10, 0, 3, 7, 21, 0, 0, 5, 9, 15, 0, 0, 3, 4, 6, 8, 12, 16, 24, 0, 0, 0, 0, 0, 3, 9, 27, 0

Offset: 1

Omar E. Pol, Dec 23 2024

The number of positive terms in row n is A174903(n).
The indices of the rows that contain a zero give A174905.
The indices of the rows that contain positive integers give A005279.
The positive integers in the n-th row are the missing divisors of n in the n-th row of A379374.
The odd integers in the n-th row are the missing odd divisors of n in the n-th row of A379288.

			Triangle begins:
  0;
  0;
  0;
  0;
  0;
  3;
  0;
  0;
  0;
  0;
  0;
  3, 4, 6;
  0;
  0;
  5;
  0;
  0;
  3, 9;
  0;
  5;
  ...
From _Omar E. Pol_, Apr 19 2025: (Start)
For n = 12 there are three divisors m of 12 such that there is a divisor d of 12 with d < m < 2*d. Those divisors are 3, 4 and 6 as shown below:
   d  <  m  <  2*d
--------------------
   1            2
   2     3      4
   3     4      6
   4     6      8
   6           12
  12           24
.
So the 12th row of the triangle is [3, 4, 6]. (End)

Cf. A005279, A027750, A174903, A174905, A182469, A237271, A379288, A379374, A379379, A379384, A383209.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1], e}, e = Select[d, #[[2]] < 2*#[[1]] &][[;; , 2]]; If[e == {}, {0}, e]]; Table[row[n], {n, 1, 55}] // Flatten (* Amiram Eldar, Dec 23 2024 *)

More terms from Amiram Eldar, Dec 23 2024

Name changed by Omar E. Pol, Feb 05 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

Omar E. Pol, Aug 23 2025

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.

			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].

Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

Row sums give A033273.

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

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

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

Omar E. Pol, Aug 23 2025

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.

			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.

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

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

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

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.

Original entry on oeis.org

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

Omar E. Pol, Apr 19 2025

			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].

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.

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 *)

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

Omar E. Pol, Aug 27 2025

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.

			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.

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

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

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

More terms from Alois P. Heinz, Aug 27 2025

cp's OEIS Frontend

A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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A379384 Sum of the divisors of n except the "e" divisors described in A005279.

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A379461 Irregular triangle read by rows in which row n lists the divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such divisors do not exist.

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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.

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

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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.

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

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Extensions