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A324976 Rank of the n-th primary Carmichael number.

%I A324976 #25 Jun 30 2022 08:37:51
%S A324976 12,8,18,12,52,52,20,32,16,54,8,36,124,34,12,72,96,26,28,76,98,1804,
%T A324976 108,124,18,72,172,120,10,104,32,244,130,376,18,92,780,36,172,92,284,
%U A324976 24,198,12,244,64,234,340,100,284,24,124,44,518,364,16,82,148,8,206
%N A324976 Rank of the n-th primary Carmichael number.
%C A324976 See A324974 for definition and explanation of rank of a special polygonal number, hence of rank of a primary Carmichael number A324316 by Kellner and Sondow 2019.
%H A324976 Amiram Eldar, <a href="/A324976/b324976.txt">Table of n, a(n) for n = 1..10000</a>
%H A324976 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 A324976 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 A324976 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>
%F A324976 a(n) = 2+2*((m/p)-1)/(p-1), where m = A324316(n) and p is its greatest prime factor. Hence a(n) is even; see Formula in A324975.
%e A324976 If m = A324316(1) = 1729 = 7*13*19, then p = 19, so a(1) = 2+2*((1729/19)-1)/(19-1) = 12.
%t A324976 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
%t A324976 LP[n_] := Transpose[FactorInteger[n]][[1]];
%t A324976 TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
%t A324976 T = Select[Range[1, 10^7, 2], TestCP[#] &];
%t A324976 GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
%t A324976 Table[2 + 2*(T[[i]]/GPF[T[[i]]] - 1)/(GPF[T[[i]]] - 1), {i, Length[T]}]
%Y A324976 Subsequence of A324975 (rank of the n-th Carmichael number A002997) and of A324974 (rank of the n-th special polygonal number A324973).
%Y A324976 Cf. also A324316, A324972.
%K A324976 nonn,base
%O A324976 1,1
%A A324976 _Bernd C. Kellner_ and _Jonathan Sondow_, Mar 24 2019
%E A324976 More terms from _Amiram Eldar_, Mar 27 2019