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A331682
One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.
6
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 62, 64, 66, 67, 68, 70, 71, 74, 76, 77, 79, 80, 82, 85, 86, 88, 89, 93, 94, 95, 96, 101
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.
Sequence in context: A328335 A302569 A235034 * A107037 A031994 A032897
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 27 2020
STATUS
approved