A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Author

Gus Wiseman, Feb 02 2020

Keywords

• nonn

Comments

First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).

First differs from A331683 in having 49.

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.

Lone-child-avoiding means there are no unary branchings.

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.

Consists of one and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of all lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
The sequence of terms together with their prime indices begins:
     1: {}                  212: {1,1,16}
     4: {1,1}               214: {1,28}
     8: {1,1,1}             224: {1,1,1,1,1,4}
    14: {1,4}               256: {1,1,1,1,1,1,1,1}
    16: {1,1,1,1}           262: {1,32}
    28: {1,1,4}             304: {1,1,1,1,8}
    32: {1,1,1,1,1}         326: {1,38}
    38: {1,8}               343: {4,4,4}
    49: {4,4}               344: {1,1,1,14}
    56: {1,1,1,4}           361: {8,8}
    64: {1,1,1,1,1,1}       392: {1,1,1,4,4}
    76: {1,1,8}             424: {1,1,1,16}
    86: {1,14}              428: {1,1,28}
    98: {1,4,4}             448: {1,1,1,1,1,1,4}
   106: {1,16}              454: {1,49}
   112: {1,1,1,1,4}         512: {1,1,1,1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}     524: {1,1,32}
   152: {1,1,1,8}           526: {1,56}
   172: {1,1,14}            608: {1,1,1,1,1,8}
   196: {1,1,4,4}           622: {1,64}
		

Links

• David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
• Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Crossrefs

Not requiring local disjointness gives A291636.

Not requiring lone-child avoidance gives A316495.

A superset of A320269.

These trees are counted by A331680.

The semi-identity tree version is A331683.

The version containing 2 is A331873.

Cf. A001678, A007097, A050381, A061775, A196050, A302569, A302696, A316473, A316694, A316696, A316697, A331682, A331686, A331687, A331872, A331935.

Programs

Mathematica

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

Formula

Intersection of A291636 and A316495.