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A355662
Smallest number of children of any vertex which has children, in the rooted tree with Matula-Goebel number n.
1
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, 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, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1
OFFSET
1,4
COMMENTS
Record highs are at a(2^k) = k which is a root with k singleton children.
If n is prime then the root has a single child so that a(n) = 1.
FORMULA
a(n) = min(bigomega(n), {a(primepi(p)) | p odd prime factor of n}).
a(n) = Min_{s>=2 in row n of A354322} bigomega(s).
EXAMPLE
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.
31972 root
/ | \
1 1 1007 Tree n=31972 and its
/ \ subtree numbers.
8 16
/|\ // \\
1 1 1 1 1 1 1
MAPLE
a:= proc(n) option remember; uses numtheory;
min(bigomega(n), map(p-> a(pi(p)), factorset(n) minus {2})[])
end:
seq(a(n), n=1..100); # Alois P. Heinz, Jul 15 2022
MATHEMATICA
a[n_] := a[n] = Min[Join[{PrimeOmega[n]}, a /@ PrimePi @ Select[ FactorInteger[n][[All, 1]], #>2&]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 08 2022 *)
PROG
(PARI) a(n) = my(f=factor(n)); vecmin(concat(vecsum(f[, 2]), [self()(primepi(p)) |p<-f[, 1], p!=2]));
CROSSREFS
Cf. A000720, A001222 (bigomega), A354322 (distinct subtrees).
Cf. A291636 (indices of !=1).
Cf. A355661 (maximum children).
Sequence in context: A095025 A274382 A318997 * A069897 A257242 A337908
KEYWORD
nonn
AUTHOR
Kevin Ryde, Jul 15 2022
STATUS
approved