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%I #14 Oct 20 2017 15:40:40
%S 1,2,6,30,78,330,390,870,1410,3198,3390,4290,7878,9570,10230,11310,
%T 13026,15510,15990,18330,26070,30966,37290,39390,40890,44070,45210,
%U 65130,84810,94830,98310,104610,122070,124410,132990,154830,159330,175890,198330,201630
%N Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).
%C A rooted tree is transitive if every terminal subtree is a branch of the root. A finitary set is transitive if every element is also a subset.
%e Let o = {}. The sequence of transitive finitary sets begins:
%e 1 o
%e 2 {o}
%e 6 {o,{o}}
%e 30 {o,{o},{{o}}}
%e 78 {o,{o},{o,{o}}}
%e 330 {o,{o},{{o}},{{{o}}}}
%e 390 {o,{o},{{o}},{o,{o}}}
%e 870 {o,{o},{{o}},{o,{{o}}}}
%e 1410 {o,{o},{{o}},{{o},{{o}}}}
%e 3198 {o,{o},{o,{o}},{{o,{o}}}}
%e 3390 {o,{o},{{o}},{o,{o},{{o}}}}
%e 4290 {o,{o},{{o}},{{{o}}},{o,{o}}}
%e 7878 {o,{o},{o,{o}},{o,{o,{o}}}}
%e 9570 {o,{o},{{o}},{{{o}}},{o,{{o}}}}
%e 10230 {o,{o},{{o}},{{{o}}},{{{{o}}}}}
%e 11310 {o,{o},{{o}},{o,{o}},{o,{{o}}}}
%e 13026 {o,{o},{o,{o}},{{o},{o,{o}}}}
%e 15510 {o,{o},{{o}},{{{o}}},{{o},{{o}}}}
%e 15990 {o,{o},{{o}},{o,{o}},{{o,{o}}}}
%e 18330 {o,{o},{{o}},{o,{o}},{{o},{{o}}}}
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t finitaryQ[n_]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
%t subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
%t transitaryQ[n_]:=Divisible[n,Times@@subprimes[n]];
%t nn=100000;Fold[Select,Range[nn],{finitaryQ,transitaryQ}]
%Y Cf. A000081, A001192, A004111, A007097, A076146, A276625, A279861, A290689, A290822.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 19 2017
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