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A331490
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Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).
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8
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8, 16, 28, 32, 56, 64, 76, 98, 112, 128, 152, 172, 196, 212, 224, 256, 266, 304, 343, 344, 392, 424, 428, 448, 512, 524, 532, 602, 608, 652, 686, 688, 722, 742, 784, 848, 856, 896, 908, 931, 1024, 1048, 1052, 1064, 1204, 1216, 1244, 1304, 1372, 1376, 1444
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OFFSET
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1,1
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COMMENTS
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We say that a rooted tree is (topologically) series-reduced if no vertex has degree 2.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.
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LINKS
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EXAMPLE
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The sequence of all series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:
8: (ooo)
16: (oooo)
28: (oo(oo))
32: (ooooo)
56: (ooo(oo))
64: (oooooo)
76: (oo(ooo))
98: (o(oo)(oo))
112: (oooo(oo))
128: (ooooooo)
152: (ooo(ooo))
172: (oo(o(oo)))
196: (oo(oo)(oo))
212: (oo(oooo))
224: (ooooo(oo))
256: (oooooooo)
266: (o(oo)(ooo))
304: (oooo(ooo))
343: ((oo)(oo)(oo))
344: (ooo(o(oo)))
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
srQ[n_]:=Or[n==1, With[{m=primeMS[n]}, And[Length[m]>1, And@@srQ/@m]]];
Select[Range[1000], PrimeOmega[#]>2&&srQ[#]&]
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CROSSREFS
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These trees are counted by A331488.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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