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

%I A324972 #25 Apr 27 2024 03:35:46
%S A324972 6,10,15,21,22,30,33,34,35,39,42,46,51,55,57,58,65,66,69,70,78,82,85,
%T A324972 87,91,93,94,95,102,105,106,111,114,115,118,123,129,130,133,138,141,
%U A324972 142,145,154,155,159,165,166,174,177,178,183,185,186,190,195,201,202
%N A324972 Squarefree polygonal numbers P(s,n) with s >= 3 and n >= 3.
%C A324972 The main entry for this sequence is A090466 = polygonal numbers of order (or rank) greater than 2.
%C A324972 The special polygonal numbers A324973 form a subsequence that contains all Carmichael numbers A002997. See Kellner and Sondow 2019.
%H A324972 Amiram Eldar, <a href="/A324972/b324972.txt">Table of n, a(n) for n = 1..10000</a>
%H A324972 Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), #A52, 21 pp.; <a href="https://arxiv.org/abs/1902.10672">arXiv preprint</a>, arXiv:1902.10672 [math.NT], 2019-2021.
%H A324972 Bernd C. Kellner, <a href="http://math.colgate.edu/~integers/w38/w38.pdf">On primary Carmichael numbers</a>, Integers 22 (2022), #A38, 39 pp.; <a href="https://arxiv.org/abs/1902.11283">arXiv preprint</a>, arXiv:1902.11283 [math.NT], 2019-2022.
%H A324972 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.
%F A324972 Squarefree P(s,n) = (n^2*(s-2)-n*(s-4))/2 with s >= 3 and n >= 3.
%e A324972 P(3,3) = 6 which is squarefree, so a(1) = 6.
%t A324972 mx = 250; n = s = 3; lst = {};
%t A324972 While[s < Floor[mx/3] + 2, a = (n^2 (s - 2) - n (s - 4))/2;
%t A324972 If[a < mx + 1, AppendTo[lst, a], (s++; n = 2)]; n++]; lst = Union@lst;
%t A324972 Select[lst, SquareFreeQ]
%o A324972 (PARI) isok(n) = if (!issquarefree(n), return (0)); for(s=3, n\3+1, ispolygonal(n, s) && return(s)); \\ _Michel Marcus_, Mar 24 2019
%Y A324972 Intersection of A005117 and A090466.
%Y A324972 Includes A324973 which contains A002997.
%K A324972 nonn
%O A324972 1,1
%A A324972 _Bernd C. Kellner_ and _Jonathan Sondow_, Mar 21 2019