A355662 - OEIS

%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