cp's OEIS Frontend

A355662 Smallest number of children of any vertex which has children, in the rooted tree with Matula-Goebel number n.

%I A355662 #12 Sep 08 2022 08:14:21
%S A355662 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 A355662 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 A355662 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,2,1
%N A355662 Smallest number of children of any vertex which has children, in the rooted tree with Matula-Goebel number n.
%C A355662 Record highs are at a(2^k) = k which is a root with k singleton children.
%C A355662 If n is prime then the root has a single child so that a(n) = 1.
%H A355662 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A355662 a(n) = min(bigomega(n), {a(primepi(p)) | p odd prime factor of n}).
%F A355662 a(n) = Min_{s>=2 in row n of A354322} bigomega(s).
%e A355662 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 A355662     31972  root
%e A355662    / |  \
%e A355662   1  1  1007      Tree n=31972 and its
%e A355662        /    \     subtree numbers.
%e A355662       8      16
%e A355662      /|\    // \\
%e A355662     1 1 1  1 1 1 1
%p A355662 a:= proc(n) option remember; uses numtheory;
%p A355662       min(bigomega(n), map(p-> a(pi(p)), factorset(n) minus {2})[])
%p A355662     end:
%p A355662 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jul 15 2022
%t A355662 a[n_] := a[n] = Min[Join[{PrimeOmega[n]}, a /@ PrimePi @ Select[ FactorInteger[n][[All, 1]], #>2&]]];
%t A355662 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Sep 08 2022 *)
%o A355662 (PARI) a(n) = my(f=factor(n)); vecmin(concat(vecsum(f[,2]), [self()(primepi(p)) |p<-f[,1], p!=2]));
%Y A355662 Cf. A000720, A001222 (bigomega), A354322 (distinct subtrees).
%Y A355662 Cf. A291636 (indices of !=1).
%Y A355662 Cf. A355661 (maximum children).
%K A355662 nonn
%O A355662 1,4
%A A355662 _Kevin Ryde_, Jul 15 2022