A331937 a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).

1, 2, 6, 26, 202, 2462, 43954, 1063462, 33076174, 1270908802, 58596709306, 3170266564862, 197764800466826, 14024066291995502, 1117378164606478094

Offset: 1

Author

Gus Wiseman, Feb 07 2020

Keywords

• nonn
• more

Comments

Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted identity trees. A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex. It is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. In an identity tree, the branches of any given vertex are all distinct. The Matula-Goebel number of a 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.

Examples

			The sequence of terms together with their associated trees begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
   202: (o(o(o(o))))
  2462: (o(o(o(o(o)))))
		

Crossrefs

The semi-identity tree version is A331681.

Not requiring an identity tree gives A331873.

Not requiring local disjointness gives A331963.

Not requiring lone-child-avoidance gives A316494.

MG-numbers of semi-lone-child-avoiding rooted trees are A331935.

Cf. A007097, A061775, A276625, A316471, A316495, A316694, A331679, A331683, A331686, A331872, A331934, A331936, A331964, A331965.

Programs

Mathematica

msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000],msiQ]

Formula

Intersection of A276625 (identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding).

Extensions

a(14)-a(15) from Giovanni Resta, Feb 10 2020