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