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Rank of the n-th Carmichael number.
5

%I #22 Jun 30 2022 08:37:47

%S 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 6,6,156,20,46,36,32,16,8,304,36,20,36,10,316,76,468,8,30,24,1580,84,

%U 54,8,12,250,28,92,36,20,418,456,928,188,16,8,276,284,56,144

%N Rank of the n-th Carmichael number.

%C 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 The ranks of the primary Carmichael numbers A324316 form the subsequence A324976.

%H Amiram Eldar, <a href="/A324975/b324975.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.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.

%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.; arXiv:<a href="https://arxiv.org/abs/1902.11283">1902.11283</a> [math.NT], 2019.

%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 = 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 If m = A002997(1) = 561 = 3*11*17, then p = 17, so a(1) = 2+2*((561/17)-1)/(17-1) = 6.

%t T = Cases[Range[1, 10000000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]];

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

%t Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]

%Y Subsequence of A324974.

%Y A324976 is a subsequence.

%Y Cf. also A002997, A324316, A324972, A324973, A324977.

%K nonn

%O 1,1

%A _Bernd C. Kellner_ and _Jonathan Sondow_, Mar 24 2019