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A324975 Rank of the n-th Carmichael number.

%I A324975 #22 Jun 30 2022 08:37:47
%S A324975 6,10,12,8,8,10,6,6,8,18,52,12,12,18,98,164,22,6,50,8,96,34,52,46,52,
%T A324975 6,6,156,20,46,36,32,16,8,304,36,20,36,10,316,76,468,8,30,24,1580,84,
%U A324975 54,8,12,250,28,92,36,20,418,456,928,188,16,8,276,284,56,144
%N A324975 Rank of the n-th Carmichael number.
%C A324975 See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a Carmichael number A002997 by Kellner and Sondow 2019.
%C A324975 The ranks of the primary Carmichael numbers A324316 form the subsequence A324976.
%H A324975 Amiram Eldar, <a href="/A324975/b324975.txt">Table of n, a(n) for n = 1..10000</a>
%H A324975 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.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.
%H A324975 Bernd C. Kellner, <a href="http://math.colgate.edu/~integers/w38/w38.pdf">On primary Carmichael numbers</a>, Integers 22 (2022), #A38, 39 pp.; arXiv:<a href="https://arxiv.org/abs/1902.11283">1902.11283</a> [math.NT], 2019.
%H A324975 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>
%F A324975 a(n) = 2+2*((m/p)-1)/(p-1), where m = A002997(n) and p is its greatest prime factor. (See Formula in A324974.) Hence a(n) is even, by Carmichael's theorem that p-1 divides (m/p)-1, for any prime factor p of a Carmichael number m.
%e A324975 If m = A002997(1) = 561 = 3*11*17, then p = 17, so a(1) = 2+2*((561/17)-1)/(17-1) = 6.
%t A324975 T = Cases[Range[1, 10000000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];
%t A324975 GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
%t A324975 Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]
%Y A324975 Subsequence of A324974.
%Y A324975 A324976 is a subsequence.
%Y A324975 Cf. also A002997, A324316, A324972, A324973, A324977.
%K A324975 nonn
%O A324975 1,1
%A A324975 _Bernd C. Kellner_ and _Jonathan Sondow_, Mar 24 2019