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%I #27 May 11 2021 06:28:03
%S 1,0,0,1,4,18,102,624,4476,36248,329890,3326054,36846276,444783906,
%T 5811885808,81729607680,1230752346352,19760412095328,336967037143578,
%U 6082255011151724,115852476579789984,2322315553090615850,48869596859895986086,1077167364116800207968
%N Number of orbits in A002619 consisting of n permutations.
%C Also, the number of aperiodic oriented cycles on n nodes up to rotation of the nodes. See A324513 for illustrations of aperiodic undirected cycles. - _Andrew Howroyd_, Aug 16 2019
%H Harry J. Smith, <a href="/A064852/b064852.txt">Table of n, a(n) for n = 1..100</a>
%F a(n) = Sum_{k|n} mu(n/k)*phi(n/k)*(n/k)^k*k!/n^2 = A047918(n, n)/n^2.
%e n=6: The orbit {(124635)(235146)(346251)(451362)(562413)(613524)} consists of 6 single permutations.
%t a[n_] := Sum[ MoebiusMu[n/k] * EulerPhi[n/k] * (n/k)^k * (k!/n^2), {k, Divisors[n]}]; Table[a[n], {n, 1, 22}] (* _Jean-François Alcover_, Jun 26 2012, after PARI *)
%o (PARI) for(n=1,23,print(sumdiv(n,d,moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2)))
%o (PARI) { for (n=1, 100, a=sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2); write("b064852.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 28 2009
%Y Cf. A000010, A002619, A008683, A047918, A324513.
%K nice,nonn
%O 1,5
%A Michael Steyer (m.steyer(AT)osram.de), Oct 06 2001
%E Corrected and extended by _Jason Earls_ and _Vladeta Jovovic_, Oct 08 2001
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