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 A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality). 13
 1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))). A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees. Consists of 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence. LINKS Table of n, a(n) for n=1..60. David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014). Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices. FORMULA Intersection of A320230 and A331935. EXAMPLE The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins: 1: o 2: (o) 4: (oo) 6: (o(o)) 8: (ooo) 9: ((o)(o)) 12: (oo(o)) 14: (o(oo)) 16: (oooo) 18: (o(o)(o)) 24: (ooo(o)) 26: (o(o(o))) 27: ((o)(o)(o)) 28: (oo(oo)) 32: (ooooo) 36: (oo(o)(o)) 38: (o(ooo)) 46: (o((o)(o))) 48: (oooo(o)) 49: ((oo)(oo)) The sequence of terms together with their prime indices begins: 1: {} 52: {1,1,6} 152: {1,1,1,8} 2: {1} 54: {1,2,2,2} 162: {1,2,2,2,2} 4: {1,1} 56: {1,1,1,4} 169: {6,6} 6: {1,2} 64: {1,1,1,1,1,1} 172: {1,1,14} 8: {1,1,1} 72: {1,1,1,2,2} 178: {1,24} 9: {2,2} 74: {1,12} 184: {1,1,1,9} 12: {1,1,2} 76: {1,1,8} 192: {1,1,1,1,1,1,2} 14: {1,4} 81: {2,2,2,2} 196: {1,1,4,4} 16: {1,1,1,1} 86: {1,14} 202: {1,26} 18: {1,2,2} 92: {1,1,9} 206: {1,27} 24: {1,1,1,2} 96: {1,1,1,1,1,2} 208: {1,1,1,1,6} 26: {1,6} 98: {1,4,4} 212: {1,1,16} 27: {2,2,2} 104: {1,1,1,6} 214: {1,28} 28: {1,1,4} 106: {1,16} 216: {1,1,1,2,2,2} 32: {1,1,1,1,1} 108: {1,1,2,2,2} 224: {1,1,1,1,1,4} 36: {1,1,2,2} 112: {1,1,1,1,4} 243: {2,2,2,2,2} 38: {1,8} 122: {1,18} 244: {1,1,18} 46: {1,9} 128: {1,1,1,1,1,1,1} 256: {1,1,1,1,1,1,1,1} 48: {1,1,1,1,2} 144: {1,1,1,1,2,2} 262: {1,32} 49: {4,4} 148: {1,1,12} 288: {1,1,1,1,1,2,2} MATHEMATICA msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n], {2, _}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n]; Select[Range[100], msQ] CROSSREFS A superset of A000079. The non-lone-child-avoiding version is A320230. The non-semi version is A320269. These trees are counted by A331933. Not requiring semi-achirality gives A331935. The fully-achiral case is A331992. Achiral trees are counted by A003238. Numbers with at most one distinct odd prime factor are A070776. Matula-Goebel numbers of achiral rooted trees are A214577. Matula-Goebel numbers of semi-identity trees are A306202. Numbers S with at most one distinct prime index in S are A331912. Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991. Sequence in context: A161819 A331935 A331873 * A364123 A244799 A347262 Adjacent sequences: A331933 A331934 A331935 * A331937 A331938 A331939 KEYWORD nonn AUTHOR Gus Wiseman, Feb 03 2020 STATUS approved

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Last modified July 16 05:19 EDT 2024. Contains 374343 sequences. (Running on oeis4.)