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A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

%I A358727 #5 Dec 02 2022 07:06:09
%S A358727 8,16,24,28,32,36,38,42,48,49,53,54,56,57,63,64,72,76,80,81,84,96,98,
%T A358727 104,106,108,112,114,120,126,128,131,133,136,140,144,147,148,152,156,
%U A358727 159,160,162,168,171,172,178,180,182,184,189,190,192,196,200,204,208
%N A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.
%C A358727 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.
%C A358727 Node-height is the number of nodes in the longest path from root to leaf.
%H A358727 Gus Wiseman, <a href="/A358727/a358727.png">The first 64 rooted trees with greater width than height.</a>
%e A358727 The terms together with their corresponding rooted trees begin:
%e A358727    8: (ooo)
%e A358727   16: (oooo)
%e A358727   24: (ooo(o))
%e A358727   28: (oo(oo))
%e A358727   32: (ooooo)
%e A358727   36: (oo(o)(o))
%e A358727   38: (o(ooo))
%e A358727   42: (o(o)(oo))
%e A358727   48: (oooo(o))
%e A358727   49: ((oo)(oo))
%e A358727   53: ((oooo))
%e A358727   54: (o(o)(o)(o))
%e A358727   56: (ooo(oo))
%e A358727   57: ((o)(ooo))
%e A358727   63: ((o)(o)(oo))
%e A358727   64: (oooooo)
%e A358727   72: (ooo(o)(o))
%e A358727   76: (oo(ooo))
%t A358727 MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358727 Select[Range[1000],Depth[MGTree[#]]-1<Count[MGTree[#],{},{-2}]&]
%Y A358727 Positions of negative terms in A358726.
%Y A358727 These trees are counted by A358728.
%Y A358727 Differences: A358580, A358724, A358726, A358729.
%Y A358727 A000081 counts rooted trees, ordered A000108.
%Y A358727 A034781 counts rooted trees by nodes and height, ordered A080936.
%Y A358727 A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358727 MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y A358727 MG core: A000040, A000720, A001222, A007097, A056239, A112798.
%Y A358727 Cf. A185650, A206487, A209638, A358576, A358577, A358578, A358587, A358730.
%K A358727 nonn
%O A358727 1,1
%A A358727 _Gus Wiseman_, Dec 01 2022