A324972 Squarefree polygonal numbers P(s,n) with s >= 3 and n >= 3.

6, 10, 15, 21, 22, 30, 33, 34, 35, 39, 42, 46, 51, 55, 57, 58, 65, 66, 69, 70, 78, 82, 85, 87, 91, 93, 94, 95, 102, 105, 106, 111, 114, 115, 118, 123, 129, 130, 133, 138, 141, 142, 145, 154, 155, 159, 165, 166, 174, 177, 178, 183, 185, 186, 190, 195, 201, 202

Offset: 1

Author

Bernd C. Kellner and Jonathan Sondow, Mar 21 2019

Keywords

• nonn

Comments

The main entry for this sequence is A090466 = polygonal numbers of order (or rank) greater than 2.

The special polygonal numbers A324973 form a subsequence that contains all Carmichael numbers A002997. See Kellner and Sondow 2019.

Examples

			P(3,3) = 6 which is squarefree, so a(1) = 6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv preprint, arXiv:1902.10672 [math.NT], 2019-2021.
• Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv preprint, arXiv:1902.11283 [math.NT], 2019-2022.
• Wikipedia, Polygonal number.

Crossrefs

Intersection of A005117 and A090466.

Includes A324973 which contains A002997.

Programs

Mathematica

mx = 250; n = s = 3; lst = {};
While[s < Floor[mx/3] + 2, a = (n^2 (s - 2) - n (s - 4))/2;
If[a < mx + 1, AppendTo[lst, a], (s++; n = 2)]; n++]; lst = Union@lst;
Select[lst, SquareFreeQ]

PARI

isok(n) = if (!issquarefree(n), return (0)); for(s=3, n\3+1, ispolygonal(n, s) && return(s)); \\ Michel Marcus, Mar 24 2019

Formula

Squarefree P(s,n) = (n^2*(s-2)-n*(s-4))/2 with s >= 3 and n >= 3.