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 A330943 Matula-Goebel numbers of singleton-reduced rooted trees. 10
 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, 49, 52, 53, 54, 56, 57, 61, 63, 64, 68, 72, 73, 74, 76, 78, 82, 84, 86, 89, 91, 96, 98, 101, 102, 104, 106, 107, 108, 111, 112, 114, 117, 119, 122, 126, 128, 129, 131 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These trees are counted by A330951. 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. 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. 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. LINKS EXAMPLE The sequence of all singleton-reduced rooted trees together with their Matula-Goebel numbers begins: 1: o 2: (o) 4: (oo) 6: (o(o)) 7: ((oo)) 8: (ooo) 12: (oo(o)) 13: ((o(o))) 14: (o(oo)) 16: (oooo) 18: (o(o)(o)) 19: ((ooo)) 21: ((o)(oo)) 24: (ooo(o)) 26: (o(o(o))) 28: (oo(oo)) 32: (ooooo) 34: (o((oo))) 36: (oo(o)(o)) 37: ((oo(o))) MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; mgsingQ[n_]:=n==1||And@@mgsingQ/@primeMS[n]&&!And@@PrimeQ/@primeMS[n]; Select[Range[100], mgsingQ] CROSSREFS The series-reduced case is A291636. Unlabeled rooted trees are counted by A000081. Numbers whose prime indices are not all prime are A330945. Singleton-reduced rooted trees are counted by A330951. Singleton-reduced phylogenetic trees are A000311. The set S of numbers whose prime indices do not all belong to S is A324694. Cf. A000669, A001678, A006450, A007097, A007821, A061775, A196050, A257994, A276625, A277098, A320628, A330944, A330948. Sequence in context: A228370 A186112 A029453 * A352089 A358458 A014855 Adjacent sequences: A330940 A330941 A330942 * A330944 A330945 A330946 KEYWORD nonn AUTHOR Gus Wiseman, Jan 13 2020 STATUS approved

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Last modified March 26 11:28 EDT 2023. Contains 361543 sequences. (Running on oeis4.)