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A358729 Difference between the number of nodes and the node-height of the rooted tree with Matula-Goebel number n.

%I A358729 #16 Oct 23 2023 15:17:34
%S A358729 0,0,0,1,0,1,1,2,2,1,0,2,1,2,2,3,1,3,2,2,3,1,2,3,3,2,4,3,1,3,0,4,2,2,
%T A358729 3,4,2,3,3,3,1,4,2,2,4,3,2,4,4,4,3,3,3,5,3,4,4,2,1,4,3,1,5,5,4,3,2,3,
%U A358729 4,4,2,5,3,3,5,4,3,4,1,4,6,2,2,5,4,3,3,3,3,5,4,4,2,3,4,5,3,5,4,5,2,4,4,4,5,4,3,6
%N A358729 Difference between the number of nodes and the node-height of the rooted tree with Matula-Goebel number n.
%C A358729 Node-height is the number of nodes in the longest path from root to leaf.
%C A358729 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 A358729 Because the number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n, i.e., A317713(n) (= 1+A324923(n)), is always at least one larger than the depth of the same tree (= A109082(n)), it follows that a(n) >= A366386(n) for all n. - _Antti Karttunen_, Oct 23 2023
%H A358729 Antti Karttunen, <a href="/A358729/b358729.txt">Table of n, a(n) for n = 1..100000</a>
%H A358729 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A358729 a(n) = A061775(n) - A358552(n).
%F A358729 a(n) = A196050(n) - A109082(n). - _Antti Karttunen_, Oct 23 2023
%e A358729 The tree (oo(oo(o))) with Matula-Goebel number 148 has 8 nodes and node-height 4, so a(148) = 4.
%t A358729 MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358729 Table[Count[MGTree[n],_,{0,Infinity}]-(Depth[MGTree[n]]-1),{n,100}]
%o A358729 (PARI)
%o A358729 A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs,t,i); pfs=factor(n); pfs[,1]=apply(t->A061775(t),pfs[,1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i,1]*pfs[i,2])}));
%o A358729 A358552(n) = { my(v=factor(n)[, 1], d=0); while(#v, d++; v=fold(setunion, apply(p->factor(primepi(p))[, 1]~, v))); (1+d); }; \\ (after program given in A109082 by _Kevin Ryde_, Sep 21 2020)
%o A358729 A358729(n) = (A061775(n)-A358552(n)); \\ _Antti Karttunen_, Oct 23 2023
%Y A358729 Positions of 0's are A007097.
%Y A358729 Positions of first appearances are A358730.
%Y A358729 Positions of 1's are A358731.
%Y A358729 Other differences: A358580, A358724, A358726.
%Y A358729 A000081 counts rooted trees, ordered A000108.
%Y A358729 A034781 counts rooted trees by nodes and height, ordered A080936.
%Y A358729 A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358729 MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y A358729 MG core: A000040, A000720, A001222, A056239, A112798.
%Y A358729 Cf. A206487, A209638, A316321, A358576, A358577, A358592, A358725, A366386.
%K A358729 nonn
%O A358729 1,8
%A A358729 _Gus Wiseman_, Nov 29 2022
%E A358729 Data section extended up to a(108) by _Antti Karttunen_, Oct 23 2023