%I #56 Apr 27 2024 03:35:39

%S 6,15,66,70,91,190,231,435,561,703,715,782,861,946,1045,1105,1426,

%T 1653,1729,1770,1785,1794,1891,2035,2278,2465,2701,2821,2926,3059,

%U 3290,3367,3486,3655,4371,4641,4830,5005,5083,5151,5365,5551,5565,5995,6441,6545,6601

%N Special polygonal numbers.

%C Squarefree polygonal numbers P(r,p) = (p^2*(r-2)-p*(r-4))/2 whose greatest prime factor is p >= 3, and whose rank (or order) is r >= 3 (see A324974).

%C The Carmichael numbers A002997 and primary Carmichael numbers A324316 are subsequences. See Kellner and Sondow 2019.

%H Amiram Eldar, <a href="/A324973/b324973.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.

%e P(3,5) = 15 is squarefree, and its greatest prime factor is 5, so 15 is a member.

%e More generally, if p is an odd prime and P(3,p) is squarefree, then P(3,p) is a member, since P(3,p) = (p^2+p)/2 = p*(p+1)/2, so p is its greatest prime factor.

%e CAUTION: P(6,7) = 91 = 7*13 is a member even though 7 is NOT its greatest prime factor, as P(6,7) = P(3,13) and 13 is its greatest prime factor.

%t GPF[n_] := Last[Select[Divisors[n], PrimeQ]];

%t T = Select[Flatten[Table[{p, (p^2*(r - 2) - p*(r - 4))/2}, {p, 3, 150}, {r, 3, 100}], 1], SquareFreeQ[Last[#]] && First[#] == GPF[Last[#]] &];

%t Take[Union[Table[Last[t], {t, T}]], 47]

%o (PARI) is(k) = if(issquarefree(k) && k>1, my(p=vecmax(factor(k)[, 1]), r); p>2 && (r=2*(k/p-1)/(p-1)) && denominator(r)==1, 0); \\ _Jinyuan Wang_, Feb 18 2021

%Y Subsequence of A324972 = intersection of A005117 and A090466.

%Y A002997, A324316, A324319 and A324320 are subsequences.

%Y Cf. A324974, A324975, A324976.

%K nonn

%O 1,1

%A _Bernd C. Kellner_ and _Jonathan Sondow_, Mar 21 2019

%E Several missing terms inserted by _Jinyuan Wang_, Feb 18 2021