A384149 -id:A384149 - cp's OEIS frontend

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

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).

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 *)

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).

A384223 Irregular triangle read by rows: T(n,k) is the sum of the k-th odd divisor and the next even divisors that are less than the next odd divisor of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 1, 3, 7, 1, 5, 3, 9, 1, 7, 15, 1, 3, 9, 3, 15, 1, 11, 3, 25, 1, 13, 3, 21, 1, 3, 5, 15, 31, 1, 17, 3, 9, 27, 1, 19, 7, 35, 1, 3, 7, 21, 3, 33, 1, 23, 3, 57, 1, 5, 25, 3, 39, 1, 3, 9, 27, 7, 49, 1, 29, 3, 3, 21, 45, 1, 31, 63, 1, 3, 11, 33, 3, 51, 1, 5, 7, 35, 3, 13, 75, 1, 37, 3, 57, 1, 3, 13, 39, 7, 83

Offset: 1

Omar E. Pol, Jun 03 2025

If n is odd the row n lists the divisors of n.

If n is a power of 2 then row n is 2*n - 1.

			Triangle begins:
   1;
   3;
   1,  3;
   7;
   1,  5;
   3,  9;
   1,  7;
  15;
   1,  3,  9;
   3, 15;
   1, 11;
   3, 25;
   1, 13;
   3, 21;
   1,  3,  5,  15;
  31;
  ...
For n = 30 the list of divisors of 30 is [1, 2, 3, 5, 6, 10, 15, 30]. There are four sublists of divisors whose first term is odd. They are [1, 2], [3], [5, 6, 10], [15, 30]. The sum of the divisors in the sublists are respectively [3, 3, 21, 45], the same as the 30th row of the triangle.

Row sums give A000203.
Row lengths give A001227.
Companion of A384224.
Cf. A000079, A027750, A237270, A237271, A237593, A279387, A384149.

row(n) = {my(d = divisors(n), res = vector(#d / (valuation(n, 2)+1)), t = 1, s = 1); for(i = 2, #d, if(bitand(d[i], 1), res[t] = s; t++; s = d[i], s += d[i])); res[#res] = s; res} \\ David A. Corneth, Jun 08 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

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

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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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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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A384223 Irregular triangle read by rows: T(n,k) is the sum of the k-th odd divisor and the next even divisors that are less than the next odd divisor of n, with n >= 1, k >= 1.

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PARI

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

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