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 A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees. 5
 1, 2, 4, 8, 9, 16, 27, 32, 49, 64, 81, 128, 243, 256, 343, 361, 512, 529, 729, 1024, 2048, 2187, 2401, 2809, 4096, 6561, 6859, 8192, 10609, 12167, 16384, 16807, 17161, 19683, 32768, 51529, 59049, 65536, 96721, 117649, 130321, 131072, 148877, 175561, 177147 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. In an achiral rooted tree, the branches of any given vertex are all equal. 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 one, two, and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence. LINKS FORMULA Intersection of A214577 (achiral) and A331935 (semi-lone-child-avoiding). EXAMPLE The sequence of all semi-lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:      1: o      2: (o)      4: (oo)      8: (ooo)      9: ((o)(o))     16: (oooo)     27: ((o)(o)(o))     32: (ooooo)     49: ((oo)(oo))     64: (oooooo)     81: ((o)(o)(o)(o))    128: (ooooooo)    243: ((o)(o)(o)(o)(o))    256: (oooooooo)    343: ((oo)(oo)(oo))    361: ((ooo)(ooo))    512: (ooooooooo)    529: (((o)(o))((o)(o)))    729: ((o)(o)(o)(o)(o)(o))   1024: (oooooooooo) MATHEMATICA msQ[n_]:=n<=2||!PrimeQ[n]&&Length[FactorInteger[n]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n]; Select[Range[10000], msQ] CROSSREFS Except for two, a subset of A025475 (nonprime prime powers). Not requiring achirality gives A331935. The semi-achiral version is A331936. The fully-chiral version is A331963. The semi-chiral version is A331994. The non-semi version is counted by A331967. The enumeration of these trees by vertices is A331991. Achiral rooted trees are counted by A003238. MG-numbers of achiral rooted trees are A214577. Cf. A001678, A007097, A050381, A061775, A167865, A196050, A276625, A280996, A291441, A291636, A320230, A320269, A331912, A331933, A331965. Sequence in context: A046678 A330606 A046680 * A113570 A065391 A161792 Adjacent sequences:  A331989 A331990 A331991 * A331993 A331994 A331995 KEYWORD nonn AUTHOR Gus Wiseman, Feb 06 2020 STATUS approved

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Last modified April 1 05:33 EDT 2020. Contains 333155 sequences. (Running on oeis4.)