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%I #12 Sep 08 2022 08:14:21
%S 0,1,1,2,1,1,1,3,1,1,1,1,1,2,1,4,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,5,1,1,
%T 1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,6,1,1,1,1,
%U 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,2,1
%N Smallest number of children of any vertex which has children, in the rooted tree with Matula-Goebel number n.
%C Record highs are at a(2^k) = k which is a root with k singleton children.
%C If n is prime then the root has a single child so that a(n) = 1.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(n) = min(bigomega(n), {a(primepi(p)) | p odd prime factor of n}).
%F a(n) = Min_{s>=2 in row n of A354322} bigomega(s).
%e For n=31972, the tree is as follows and vertex 1007 has 2 children which is the least among the vertices which have children, so a(31972) = 2.
%e 31972 root
%e / | \
%e 1 1 1007 Tree n=31972 and its
%e / \ subtree numbers.
%e 8 16
%e /|\ // \\
%e 1 1 1 1 1 1 1
%p a:= proc(n) option remember; uses numtheory;
%p min(bigomega(n), map(p-> a(pi(p)), factorset(n) minus {2})[])
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jul 15 2022
%t a[n_] := a[n] = Min[Join[{PrimeOmega[n]}, a /@ PrimePi @ Select[ FactorInteger[n][[All, 1]], #>2&]]];
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Sep 08 2022 *)
%o (PARI) a(n) = my(f=factor(n)); vecmin(concat(vecsum(f[,2]), [self()(primepi(p)) |p<-f[,1], p!=2]));
%Y Cf. A000720, A001222 (bigomega), A354322 (distinct subtrees).
%Y Cf. A291636 (indices of !=1).
%Y Cf. A355661 (maximum children).
%K nonn
%O 1,4
%A _Kevin Ryde_, Jul 15 2022
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