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%I #28 Jun 12 2022 06:45:36
%S 0,0,0,0,0,0,0,1,2,7,12,31,58,126,233,484,904,1800,3395,6643,12612,
%T 24457,46655,90157,172750,333498,641214,1238664,2388618,4620006,
%U 8931536,17302033,33521792,65042495,126257160,245361171,477087772,928510506,1808145395,3523813566
%N Number of chiral pairs of set partitions of a primitive cycle of n elements having exactly two different elements.
%F a(n) = ((1/(2n)) * Sum_{odd d|n} mu(d)*2^(n/d) - Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2, where mu is the Möbius function at A008683.
%F a(n) = A000048(n) - A000046(n) = (A000048(n) - A179781(n))/2 = A000046(n) - A179781(n).
%F A059053(n) = Sum_{d|n} a(d).
%e For a(7)=1, the chiral pair is 0001011-0001101. For a(8)=2, the chiral pairs are 00001011-00001101 and 00010011-00011001.
%t Join[{0}, Table[(DivisorSum[NestWhile[#/2 &, n, EvenQ], MoebiusMu[#] 2^(n/#) &]/(2 n) - DivisorSum[n, MoebiusMu[n/#] 2^Floor[#/2] &])/2, {n, 1, 40}]]
%o (PARI) a(n) = if (n, (sumdiv(n, d, if (d%2, moebius(d)*2^(n/d)))/(2*n) - sumdiv(n, d, moebius(n/d)*2^(d\2)))/2, 0); \\ _Michel Marcus_, Jun 27 2019; corrected Jun 12 2022
%Y Cf. A000048 (oriented), A000046 (unoriented), A179781 (achiral), A059053 (not primitive).
%K nonn
%O 0,9
%A _Robert A. Russell_, Jun 18 2019
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