A386999 Positive 2-dense triangular numbers.

1, 6, 28, 36, 66, 120, 210, 276, 300, 378, 496, 528, 630, 780, 990, 1128, 1176, 1540, 1596, 1770, 2016, 2080, 2556, 2850, 3160, 3240, 3486, 3570, 3828, 4560, 4950, 5460, 5778, 6216, 6786, 7140, 7260, 8128, 8256, 8646, 8778, 9180, 9870, 10296, 10440, 10878, 11628

Offset

1,2

Comments

Triangular A000217 whose divisors increase by a factor of at most 2.

Also these could be called 2-densely divisible triangular numbers (see A174973).

Also triangular numbers whose symmetric representation of sigma is a polygon. For the relationship between every edge of the polygon and the Dyck paths and the partitions into consecutive parts see A235791 and A237591.

Also numbers k with the property that the symmetric representation of sigma(k) is a polygon and on the axis of symmetry of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak (in this case k is also a 2-dense hexagonal number), or vice versa, the smallest Dyck path has a peak and the largest Dyck path has a valley (in this case k is also a 2-dense second hexagonal number).

All terms have at least a middle divisor.

All even perfect numbers are terms.

Links

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

Example

6 is a triangular number and its divisors are [1, 2, 3, 6]. The divisors of 6 increase by factors of at most 2, so 6 is in the sequence.
On the other hand the diagram of the symmetric representation of sigma(6) is formed by two Dyck paths described in the rows 5 and 6 of the triangle A237593, they are [3, 2, 2, 3] and [4, 1, 1, 1, 1, 4] respectively. Both Dyck paths never touch, therefore the diagram is a polygon, more precisely it is a concave 12-gon of area A000203(6) = 12 as shown below, so 6 is in the sequence.
   y
   .
   ._ _ _ _
   |_ _ _  |_
   .     |   |_
   .     |_ _  |
   .         | |
   .         | |
   . . . . . |_| . . x
.
Also 6 has the property that the diagram of the symmetric representation of sigma(6) is a polygon and on its axis of symmetry the smallest Dyck path has a valley and the largest Dyck path has a peak (therefore 6 is also a 2-dense hexagonal number), so 6 is in the sequence.

Mathematica

Select[PolygonalNumber[Range[200]], Length[Split[Divisors[#], #2 <= 2*# &]] == 1 &] (* Paolo Xausa, Sep 06 2025 *)

Crossrefs

Intersection of A000217 and A174973 (2-dense numbers).

Cf. A000203, A000384, A000396, A014105, A067742, A235791, A237270, A237271, A237591, A237593, A262626, A346873, A380328, A384928, A386996.

Sequence in context: A252234 A334405 A376997 * A242344 A344588 A247111

Adjacent sequences: A386996 A386997 A386998 * A387000 A387001 A387002

Keyword

nonn

Author

Omar E. Pol, Aug 30 2025

Extensions

More terms from Alois P. Heinz, Aug 31 2025

Status

approved