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A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.

%I A306203 #5 Jan 30 2019 00:01:32
%S A306203 1,2,3,4,5,7,8,11,16,17,19,21,31,32,53,57,59,64,67,73,85,127,128,131,
%T A306203 133,159,241,256,269,277,311,331,335,365,367,371,393,399,439,512,649,
%U A306203 709,719,739,751,917,933,937,1007,1024,1113,1139,1205,1241,1345,1523
%N A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.
%C A306203 A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root. The only balanced rooted identity trees are rooted paths.
%e A306203 The sequence of all unlabeled balanced rooted semi-identity trees together with their Matula-Goebel numbers begins:
%e A306203    1: o
%e A306203    2: (o)
%e A306203    3: ((o))
%e A306203    4: (oo)
%e A306203    5: (((o)))
%e A306203    7: ((oo))
%e A306203    8: (ooo)
%e A306203   11: ((((o))))
%e A306203   16: (oooo)
%e A306203   17: (((oo)))
%e A306203   19: ((ooo))
%e A306203   21: ((o)(oo))
%e A306203   31: (((((o)))))
%e A306203   32: (ooooo)
%e A306203   53: ((oooo))
%e A306203   57: ((o)(ooo))
%e A306203   59: ((((oo))))
%e A306203   64: (oooooo)
%e A306203   67: (((ooo)))
%e A306203   73: (((o)(oo)))
%e A306203   85: (((o))((oo)))
%t A306203 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A306203 psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n],1],And@@psidQ/@primeMS[n]];
%t A306203 mgtree[n_]:=If[n==1,{},mgtree/@primeMS[n]];
%t A306203 Select[Range[100],And[psidQ[#],SameQ@@Length/@Position[mgtree[#],{}]]&]
%Y A306203 Cf. A000081, A004111, A007097, A048816, A184155, A276625, A306200, A306201, A306202, A316467, A317710.
%K A306203 nonn
%O A306203 1,2
%A A306203 _Gus Wiseman_, Jan 29 2019