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A319225
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Number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the prime indices of n.
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16
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1, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 4, 6, 5, 5, 1, 7, 5, 8, 5, 6, 6, 9, 5, 3, 7, 2, 6, 10, 12, 11, 1, 7, 8, 7, 9, 12, 9, 8, 6, 13, 14, 14, 7, 7, 10, 15, 6, 4, 7, 9, 8, 16, 7, 8, 7, 10, 11, 17, 21, 18, 12, 8, 1, 9, 16, 19, 9, 11, 16, 20, 14, 21, 13, 8, 10, 9, 18
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OFFSET
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1,3
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COMMENTS
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a(1) = 1 by convention.
A prime index of n is a number m such that prime(m) divides n.
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LINKS
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FORMULA
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a(n) = A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.
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EXAMPLE
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Of the cycle ({1,2,3}, {(1,2),(2,3),(3,1)}) the spanning subgraphs where the sizes of connected components are (2,1) are: ({1,2,3}, {(1,2)}), ({1,2,3}, {(2,3)}), ({1,2,3}, {(3,1)}). Since the prime indices of 6 are (2,1), we conclude a(6) = 3.
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MATHEMATICA
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csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[With[{m=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]}, Select[Subsets[Partition[Range[Total[m]], 2, 1, 1], {Total[m]-PrimeOmega[n]}], Sort[Length/@csm[Union[#, List/@Range[Total[m]]]]]==m&]]], {n, 30}]
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CROSSREFS
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Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A215366, A318762, A319191, A319193, A319226.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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