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A355661 Largest number of children of any vertex in the rooted tree with Matula-Goebel number n.

%I A355661 #14 Jul 15 2022 02:37:50
%S A355661 0,1,1,2,1,2,2,3,2,2,1,3,2,2,2,4,2,3,3,3,2,2,2,4,2,2,3,3,2,3,1,5,2,2,
%T A355661 2,4,3,3,2,4,2,3,2,3,3,2,2,5,2,3,2,3,4,4,2,4,3,2,2,4,3,2,3,6,2,3,3,3,
%U A355661 2,3,3,5,2,3,3,3,2,3,2,5,4,2,2,4,2,2,2
%N A355661 Largest number of children of any vertex in the rooted tree with Matula-Goebel number n.
%C A355661 Record highs are at a(2^k) = k which is a root with k singleton children.
%C A355661 A new root above a tree has a single child (the old root) so no change to the largest number of children, except when above a singleton, so that a(prime(n)) = a(n) for n >= 2.
%C A355661 Terms a(n) <= 1 are paths down (all vertices 0 or 1 children), which are the primeth recurrence n = A007097.
%H A355661 Kevin Ryde, <a href="/A355661/b355661.txt">Table of n, a(n) for n = 1..10000</a>
%H A355661 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A355661 a(n) = max(bigomega(n), {a(primepi(p)) | p prime factor of n}).
%F A355661 a(n) = Max_{s in row n of A354322} bigomega(s).
%e A355661 For n=629, tree 629 is as follows and vertex 12 has 3 children which is the most of any vertex so that a(629) = 3.
%e A355661       629  root
%e A355661      /   \
%e A355661     7     12     tree n=629 and its
%e A355661     |     /|\    subtree numbers
%e A355661     4    1 1 2
%e A355661    / \       |
%e A355661   1   1      1
%p A355661 a:= proc(n) option remember; uses numtheory;
%p A355661       max(bigomega(n), map(p-> a(pi(p)), factorset(n))[])
%p A355661     end:
%p A355661 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jul 14 2022
%t A355661 nn = 105; a[1] = 0; a[2] = 1; Do[a[n] = Max@ Append[Map[a[PrimePi[#]] &, FactorInteger[n][[All, 1]]], PrimeOmega[n]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, Jul 14 2022 *)
%o A355661 (PARI) a(n) = my(f=factor(n)); vecmax(concat(vecsum(f[,2]), [self()(primepi(p)) |p<-f[,1]]));
%Y A355661 Cf. A001222 (bigomega), A354322 (distinct subtrees).
%Y A355661 Cf. A007097 (indices of <=1).
%Y A355661 Cf. A355662 (minimum children).
%K A355661 nonn
%O A355661 1,4
%A A355661 _Kevin Ryde_, Jul 14 2022