OFFSET
1,2
COMMENTS
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.
Also Matula-Goebel numbers of locally disjoint rooted semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. 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 a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.
EXAMPLE
The sequence of all locally disjoint rooted semi-identity trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
6: (o(o))
7: ((oo))
8: (ooo)
10: (o((o)))
11: ((((o))))
12: (oo(o))
13: ((o(o)))
14: (o(oo))
15: ((o)((o)))
16: (oooo)
17: (((oo)))
19: ((ooo))
20: (oo((o)))
22: (o(((o))))
24: (ooo(o))
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
deQ[n_]:=n==1||PrimeQ[n]&&deQ[PrimePi[n]]||CoprimeQ@@primeMS[n]&&And@@deQ/@primeMS[n];
Select[Range[100], deQ]
CROSSREFS
The non-semi identity tree case is A316494.
The enumeration of these trees by vertices is A331783.
Semi-identity trees are counted by A306200.
Matula-Goebel numbers of semi-identity trees are A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 27 2020
STATUS
approved