OFFSET
1,2
COMMENTS
Definition: A positive integer belongs to the sequence iff it is 1, 4, or a squarefree semiprime whose prime indices both already belong to the sequence. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
In a semi-identity tree, only the non-leaf branches of any given vertex are distinct. Alternatively, a rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees.
The Matula-Goebel number of an unlabeled rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
LINKS
EXAMPLE
The sequence of terms together with the corresponding unlabeled rooted trees begins:
1: o
4: (oo)
14: (o(oo))
86: (o(o(oo)))
301: ((oo)(o(oo)))
886: (o(o(o(oo))))
3101: ((oo)(o(o(oo))))
3986: (o((oo)(o(oo))))
13766: (o(o(o(o(oo)))))
13951: ((oo)((oo)(o(oo))))
19049: ((o(oo))(o(o(oo))))
48181: ((oo)(o(o(o(oo)))))
57026: (o((oo)(o(o(oo)))))
75266: (o(o((oo)(o(oo)))))
85699: ((o(oo))((oo)(o(oo))))
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mgbiQ[n_]:=Or[n==1, n==4, SquareFreeQ[n]&&PrimeOmega[n]==2&&And@@mgbiQ/@primeMS[n]];
Select[Range[1000], mgbiQ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 14 2021
STATUS
approved