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%I #34 Jul 11 2019 02:14:11
%S 1,2,3,5,11,23,47,283,719,1439,2879,34549,138197,531441,1594323,
%T 4782969,14348907,43046721,86093443,258280327,688747547
%N Smallest integer k such that the number of iterations of Carmichael lambda function (A002322) needed to reach 1 starting at k (k is counted) is n.
%C Smallest number k such that the trajectory of k under iteration of Carmichael lambda function contains exactly n distinct numbers (including k and the fixed point).
%C The first 13 terms are 1 or a prime. The next five terms are powers of 3. Then another prime. What explains this behavior? - _T. D. Noe_, Mar 23 2012
%C A185816(a(n) = n - 1. - _Reinhard Zumkeller_, Sep 02 2014
%C If a(n) (n > 1) is either a prime or a power of 3, then a(n) is also the smallest integer k such that the number of iterations of Euler's totient function (A000010) needed to reach 1 starting at k (k is counted) is n. - _Jianing Song_, Jul 10 2019
%H Nick Harland, <a href="http://arxiv.org/abs/1203.4791">The number of iterates of the Carmichael lambda function required to reach 1</a>, arXiv:1203.4791v1 [math.NT], Mar 21 2012.
%e for n=5, a(5)=11 gives a chain of length 5 because the trajectory is 11 -> 10 -> 4 -> 2 -> 1.
%t f[n_] := Length@ NestWhileList[ CarmichaelLambda, n, Unequal, 2] - 1; t = Table[0, {30}]; k = 1; While[k < 2100000001, a = f@ k; If[ t[[a]] == 0, t[[a]] = k; Print[a, " = ", k]]; k++] (* slightly modified by _Robert G. Wilson v_, Sep 01 2014 *)
%o (Haskell)
%o import Data.List (elemIndex); import Data.Maybe (fromJust)
%o a173927 = (+ 1) . fromJust . (`elemIndex` map (+ 1) a185816_list)
%o -- _Reinhard Zumkeller_, Sep 02 2014
%Y Cf. A002322, A027763, A056637.
%Y Cf. A185816 (number of iterations of Carmichael lambda function needed to reach 1), A003434 (number of iterations of Euler's totient function needed to reach 1).
%K nonn,more
%O 1,2
%A _Michel Lagneau_, Nov 26 2010
%E a(20)-a(21) from _Robert G. Wilson v_, Sep 01 2014
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