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%I #13 Jan 15 2020 00:25:24
%S 1,2,4,6,7,8,12,13,14,16,18,19,21,24,26,28,32,34,36,37,38,39,42,43,48,
%T 49,52,53,54,56,57,61,63,64,68,72,73,74,76,78,82,84,86,89,91,96,98,
%U 101,102,104,106,107,108,111,112,114,117,119,122,126,128,129,131
%N Matula-Goebel numbers of singleton-reduced rooted trees.
%C These trees are counted by A330951.
%C A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.
%C The Matula-Goebel number of a rooted tree is the product of primes of the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
%C A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is 1 or its prime indices all belong to this sequence but are not all prime.
%e The sequence of all singleton-reduced rooted trees together with their Matula-Goebel numbers begins:
%e 1: o
%e 2: (o)
%e 4: (oo)
%e 6: (o(o))
%e 7: ((oo))
%e 8: (ooo)
%e 12: (oo(o))
%e 13: ((o(o)))
%e 14: (o(oo))
%e 16: (oooo)
%e 18: (o(o)(o))
%e 19: ((ooo))
%e 21: ((o)(oo))
%e 24: (ooo(o))
%e 26: (o(o(o)))
%e 28: (oo(oo))
%e 32: (ooooo)
%e 34: (o((oo)))
%e 36: (oo(o)(o))
%e 37: ((oo(o)))
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t mgsingQ[n_]:=n==1||And@@mgsingQ/@primeMS[n]&&!And@@PrimeQ/@primeMS[n];
%t Select[Range[100],mgsingQ]
%Y The series-reduced case is A291636.
%Y Unlabeled rooted trees are counted by A000081.
%Y Numbers whose prime indices are not all prime are A330945.
%Y Singleton-reduced rooted trees are counted by A330951.
%Y Singleton-reduced phylogenetic trees are A000311.
%Y The set S of numbers whose prime indices do not all belong to S is A324694.
%Y Cf. A000669, A001678, A006450, A007097, A007821, A061775, A196050, A257994, A276625, A277098, A320628, A330944, A330948.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 13 2020
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