Omar E. Pol - cp's OEIS frontend

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

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

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).

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).

A387001 Number of vertices in the diagram called "symmetric representation of sigma(n)" where its "parts" or polygons are dissected into unit squares (see the example).

Original entry on oeis.org

4, 8, 11, 16, 17, 25, 23, 32, 32, 39, 35, 53, 41, 53, 55, 64, 53, 76, 59, 83, 75, 81, 71, 109, 82, 95, 95, 113, 89, 133, 95, 128, 115, 123, 119, 164, 113, 137, 135, 171, 125, 181, 131, 173, 169, 165, 143, 221, 156, 194, 175, 203, 161, 229, 183, 233, 195, 207, 179, 289, 185, 221, 231, 256

Offset: 1

Omar E. Pol, Aug 14 2025

Consider here that in the diagram every edge has length 1 and every face is a unit square.
The number of faces is A000203(n).
The number of edges is 2*A155085(n).
The number of edges with the same orientation is A155085(n).

			For n = 5 the diagram is as shown below:
   _ _ _
  |_|_|_|
        |_ _
          |_|
          |_|
          |_|
.
The number of vertices is a(5) = 17.
The number of faces is A000203(5) = 6.
The number of edges is 2*A155085(5) = 2*11 = 22.
The number of edges with the same orientation is A155085(5) = 11.

Omar E. Pol, Illustration of initial terms of A000203 in a concave polyhedron (n = 1..16)

Cf. A000203, A005408, A155085, A224880, A237270, A237271, A237593.

a(n) = A000203(n) + A005408(n).
a(n) = 2*A155085(n) - A000203(n) + 1. (Euler's formula: V = E - F + 1).
a(n) = A224880(n) + 1.

A387030 Irregular triangle read by rows: T(n,k) is the number of primes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 3, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 3, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1

Offset: 1

Omar E. Pol, Aug 13 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:
  0;
  1;
  0, 1;
  1;
  0, 1;
  2;
  0, 1;
  1;
  0, 1, 0;
  1, 1;
  0, 1;
  2;
  0, 1;
  1, 1;
  0, 2, 0;
  ...
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 prime 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 second sublist contains primes, so row 15 is [0, 2, 0].

Paolo Xausa, Table of n, a(n) for n = 1..12242 (rows 1..4000 of triangle, flattened).

Row sums give A001221.

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

A387030row[n_] := Map[Count[#, _?PrimeQ] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A387030row, 50] (* Paolo Xausa, Aug 19 2025 *)

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.

Original entry on oeis.org

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 *)

A386984 Number of 2-dense sublists of divisors of the n-th hexagonal number.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 3, 5, 1, 5, 1, 5, 1, 5, 1, 3, 1, 7, 3, 3, 1, 5, 3, 7, 1, 5, 1, 3, 1, 7, 3, 5, 1, 3, 1, 5, 1, 7, 1, 7, 1, 5, 3, 5, 1, 5, 1, 7, 3, 5, 1, 7, 1, 7, 5, 7, 1, 5, 3, 3, 1, 7, 1, 7, 1, 7, 5, 5, 1, 7, 1, 7, 3, 5, 1, 3, 1, 9, 3, 7, 1, 7, 1, 7, 1, 5, 1, 5, 1, 9, 3, 3, 1, 3, 1, 9, 1

Offset: 0

Omar E. Pol, Aug 11 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.
Conjecture: all terms are odd.

			For n = 3 the third positive hexagonal 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(3) = 3.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Bisection of A384928.

Cf. A000384, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386989.

A386984[n_] := Length[Split[Divisors[PolygonalNumber[6, n]], #2 <= 2*# &]];
Array[A386984, 100, 0] (* Paolo Xausa, Aug 29 2025 *)

a(n) = A237271(A000384(n)) for n >= 1 (conjectured).

A384998 Total number of partitions of all numbers <= n with designated summands, n >= 0.

Original entry on oeis.org

1, 2, 5, 10, 20, 35, 63, 104, 173, 275, 435, 666, 1018, 1516, 2248, 3275, 4745, 6776, 9632, 13528, 18910, 26182, 36078, 49311, 67111, 90690, 122052, 163271, 217559, 288350, 380806, 500504, 655601, 855113, 1111777, 1439911, 1859347, 2392509, 3069921, 3926494

Offset: 0

Omar E. Pol, Aug 06 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Partial sums of A077285.
Row sums of A384999.
Cf. A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j, j=1..n/i)))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..41);  # Alois P. Heinz, Aug 06 2025

nmax = 50; CoefficientList[Series[1/(1-x) * Product[(1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 08 2025 *)

From Vaclav Kotesovec, Aug 08 2025: (Start)
a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (2^(9/4) * sqrt(3) * Pi * n^(3/4)).
G.f.: 1/(1-x) * Product_{k>=1} (1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))). (End)

A384928 Number of 2-dense sublists of divisors of the n-th triangular number.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 1, 3, 4, 1, 2, 3, 4, 1, 2, 5, 6, 3, 1, 5, 4, 1, 1, 5, 6, 1, 4, 5, 6, 1, 1, 5, 6, 1, 2, 3, 6, 1, 2, 7, 8, 3, 1, 3, 4, 1, 1, 5, 6, 3, 4, 7, 3, 1, 1, 5, 4, 1, 2, 3, 8, 1, 1, 7, 8, 3, 3, 5, 6, 1, 2, 3, 6, 1, 4, 5, 8, 1, 1, 7, 4, 1, 1, 7, 6, 1, 4, 5, 3, 3, 3, 5, 8, 1, 2, 5, 5, 1, 6

Offset: 0

Omar E. Pol, Aug 08 2025

nonn

By definition a(n) is also the number of 2-dense sublists of divisors of the n-th generalized hexagonal number.
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.
Conjecture: all odd indexed terms are odd.

			For n = 5 the 5th triangular 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(5) = 3.
For n = 12 the 12th triangular number is 78. The list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78], so a(12) = 2. Note that 78 is also the first practical number A005153 not in the sequence of the 2-dense numbers A174973.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000217, A005153, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386984 (a bisection), A386989.

A384928[n_] := Length[Split[Divisors[PolygonalNumber[n]], #2 <= 2*# &]];
Array[A384928, 100, 0] (* Paolo Xausa, Aug 14 2025 *)

a(n) = A237271(A000217(n)) for n >= 1 (conjectured).

A384931 Number of 2-dense sublists of divisors of the number of partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 2, 1, 1, 1, 3, 2, 3, 1, 5, 6, 4, 4, 5, 1, 2, 4, 3, 4, 1, 5, 4, 7, 2, 4, 9, 10, 4, 9, 2, 6, 9, 3, 1, 9, 4, 11, 8, 4, 3, 3, 8, 12, 4, 11, 7, 10, 5, 3, 7, 2, 2, 1, 8, 5, 6, 8, 5, 2, 1, 3, 10, 6, 1, 6, 8, 7, 1, 1, 4, 2, 7, 9, 3, 4, 9, 6, 2

Offset: 0

Omar E. Pol, Jul 30 2025

In a 2-dense 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 = 7 the number of partitions of 7 is A000041(7) = 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(7) = 3.
For n = 19 the number of partitions of 19 is A000041(19) = 490. The list of divisors of 490 is [1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]. There are four 2-dense sublists of divisors of 490, they are [1, 2], [5, 7, 10, 14], [35, 49, 70, 98], [245, 490], so a(19) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A000041, A174973 (2-dense numbers), A237271, A384149, A384222, A384225, A384226, A384930.

A384931[n_] := Length[Split[Divisors[PartitionsP[n]], #2 <= 2*# &]];
Array[A384931, 100, 0] (* Paolo Xausa, Aug 28 2025 *)

a(n) = A237271(A000041(n)). Conjectured.

More terms from Alois P. Heinz, Jul 30 2025

cp's OEIS Frontend

User: Omar E. Pol

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

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

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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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A387001 Number of vertices in the diagram called "symmetric representation of sigma(n)" where its "parts" or polygons are dissected into unit squares (see the example).

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A387030 Irregular triangle read by rows: T(n,k) is the number of primes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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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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A386984 Number of 2-dense sublists of divisors of the n-th hexagonal number.

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A384998 Total number of partitions of all numbers <= n with designated summands, n >= 0.

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