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A342507
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Number of internal nodes in rooted tree with Matula-Goebel number n.
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28
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0, 1, 2, 1, 3, 2, 2, 1, 3, 3, 4, 2, 3, 2, 4, 1, 3, 3, 2, 3, 3, 4, 4, 2, 5, 3, 4, 2, 4, 4, 5, 1, 5, 3, 4, 3, 3, 2, 4, 3, 4, 3, 3, 4, 5, 4, 5, 2, 3, 5, 4, 3, 2, 4, 6, 2, 3, 4, 4, 4, 4, 5, 4, 1, 5, 5, 3, 3, 5, 4, 4, 3, 4, 3, 6, 2, 5, 4, 5, 3, 5, 4, 5, 3, 5, 3, 5, 4, 3, 5, 4, 4, 6, 5, 4, 2, 6, 3, 6, 5
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OFFSET
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1,3
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COMMENTS
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The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product_{T_i} prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration.)
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LINKS
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FORMULA
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EXAMPLE
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a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.
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MATHEMATICA
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MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Count[MGTree[n], _[__], {0, Infinity}], {n, 100}] (* Gus Wiseman, Nov 28 2022 *)
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PROG
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(PARI) A342507(n) = if( n==1, 0, my(f=factor(n)); 1+sum(k=1, matsize(f)[1], A342507(primepi(f[k, 1]))*f[k, 2]));
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CROSSREFS
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Positions of first appearances are A358554.
A358575 counts rooted trees by nodes and internals.
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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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