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%I #9 Sep 11 2018 21:50:48
%S 1,2,3,2,4,3,5,3,3,4,6,4,4,5,7,2,5,5,6,8,3,6,6,7,4,9,5,4,7,7,8,4,5,10,
%T 6,3,5,8,8,9,5,6,11,7,4,6,9,9,5,10,6,7,12,8,5,7,10,10,6,11,7,8,13,3,9,
%U 6,8,11,11,7,12,8,9,14,4,10,7,9,12,12,3,8
%N Number of antichain covers of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
%C If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) (as can be represented in functional programming languages such as Mathematica) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). The a(n) is the number of ways to partition e(n) into disjoint subexpressions such that all leaves are covered by exactly one of them.
%F If n = rad(x)^(Product_i prime(y_i)^z_i) where rad = A007916 then a(n) = 1 + a(x) * Product_i a(y_i)^z_i.
%e 441 is the e-number of o[o,o][o] which has antichain covers {o[o,o][o]}, {o[o,o], o}, {o, o, o, o}}, corresponding to the leaf-colorings 1[1,1][1], 1[1,1][2], 1[2,3][4], so a(441) = 3.
%t nn=20000;
%t radQ[n_]:=If[n==1,False,GCD@@FactorInteger[n][[All,2]]==1];
%t rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
%t Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
%t a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},1+a[radPi[n^(1/g)]]*Product[a[PrimePi[pr[[1]]]]^pr[[2]],{pr,If[g==1,{},FactorInteger[g]]}]]];
%t Array[a,100]
%Y Cf. A000081, A007853, A007916, A052409, A052410, A277576, A277996.
%Y Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 19 2018
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