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%I #37 Apr 27 2024 03:36:15
%S 3,3,3,5,3,3,6,3,6,3,11,5,3,3,8,10,5,6,12,3,15,9,3,5,3,8,3,8,19,14,5,
%T 7,3,6,6,36,21,66,22,3,10,5,6,3,3,50,10,20,5,14,11,51,3,10,21,6,13,5,
%U 16,25,3,3,6,6,12,14,10,68,5,28,3,11,29,3,56,6,19
%N Rank of the n-th special polygonal number A324973(n).
%C While two polygonal numbers of different ranks can be equal (e.g., P(6,n) = P(3,2n-1)), that cannot occur for special polygonal numbers, since for fixed p the value of P(r,p) is strictly increasing with r. Thus the rank of a special polygonal number is well-defined.
%C The Carmichael numbers A002997 and primary Carmichael numbers A324316 are special polygonal numbers (see Kellner and Sondow 2019). Their ranks form the subsequences A324975 and A324976.
%H Amiram Eldar, <a href="/A324974/b324974.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>.
%F a(n) = 2 + 2*((m/p)-1)/(p-1), where m = A324973(n) and p is its greatest prime factor. (Proof: solve m = P(r,p) = (p^2*(r-2) - p*(r-4))/2 for r.)
%e If m = A324973(4) = 70 = 2*5*7, then p = 7, so a(4) = 2+2*((70/7)-1)/(7-1) = 5.
%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 TT = Take[Union[Table[Last[T[[i]]], {i, Length[T]}]], 47];
%t Table[2 + 2*(t/GPF[t] - 1)/(GPF[t] - 1), {t, TT}]
%Y A324975 and A324976 are subsequences.
%Y Cf. A002997, A324316, A324972, A324973, A324977.
%K nonn,changed
%O 1,1
%A _Bernd C. Kellner_ and _Jonathan Sondow_, Mar 24 2019
%E Several missing terms inserted by and more terms from _Jinyuan Wang_, Feb 18 2021
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