A331682 -id:A331682 - cp's OEIS frontend

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 152, 172, 212, 214, 224, 256, 262, 304, 326, 344, 424, 428, 448, 512, 524, 526, 608, 622, 652, 688, 766, 848, 856, 886, 896, 1024, 1048, 1052, 1154, 1216, 1226, 1244, 1304, 1376, 1438, 1532, 1696

Offset: 1

Gus Wiseman, Jan 30 2020

nonn

Also Matula-Goebel numbers of lone-child-avoiding rooted trees at with at most one non-leaf branch under any given vertex. A rooted tree is lone-child-avoiding if 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 the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Also Matula-Goebel numbers of lone-child-avoiding locally disjoint semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct.

			The sequence of all lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex 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))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
  224: (ooooo(oo))

Robert Israel, Table of n, a(n) for n = 1..3140

These trees counted by number of vertices are A212804.
The semi-lone-child-avoiding version is A331681.
The non-semi-identity version is A331871.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Unlabeled semi-identity trees are counted by A306200, with Matula-Goebel numbers A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
Cf. A000081, A007097, A061775, A196050, A276625, A316470, A316696, A331678, A331679, A331680, A331682, A331686, A331687.

N:= 10^4: # for terms <= N
S:= {1}:
with(queue):
Q:= new(1):
while not empty(Q) do
  r:= dequeue(Q);
  p:= ithprime(r);
  newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;
  S:= S union newS;
  for s in newS do enqueue(Q,s) od:
od:
sort(convert(S,list)); # Robert Israel, Feb 05 2020

uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,},{p,1}}/;uryQ[PrimePi[p]])|({{2,k_}}/;k>1)];
Select[Range[100],uryQ]

Intersection of A291636, A316495, and A306202.

A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 69, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 138, 144, 148, 152, 161, 162, 169, 172, 178, 184, 192, 196, 202, 206, 207, 208, 212, 214, 216, 224, 243

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A331936 in having 69, the Matula-Goebel number of the tree ((o)((o)(o))).
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
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, two, 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.

			The sequence of all semi-lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))

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.

Not requiring lone-child-avoidance gives A316495.
A superset of A320269.
The semi-identity tree case is A331681.
The non-semi version (i.e., not containing 2) is A331871.
These trees counted by vertices are A331872.
These trees counted by leaves are A331874.
Not requiring local disjointness gives A331935.
The identity tree case is A331937.
Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.

msQ[n_]:=n==1||n==2||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 38, 48, 52, 56, 64, 74, 76, 86, 96, 104, 106, 112, 128, 148, 152, 172, 178, 192, 202, 208, 212, 214, 224, 256, 262, 296, 304, 326, 344, 356, 384, 404, 416, 424, 428, 446, 448, 478, 512, 524, 526, 592, 608, 622, 652

Offset: 1

Gus Wiseman, Jan 26 2020

nonn

Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted semi-identity trees. A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct. Note that these conditions together imply that there is at most one non-leaf branch under any given vertex.
Also Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex.
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.

			The sequence of all semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex, together with their Matula-Goebel numbers, begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  38: (o(ooo))
  48: (oooo(o))
  52: (oo(o(o)))
  56: (ooo(oo))
  64: (oooooo)
  74: (o(oo(o)))
  76: (oo(ooo))
  86: (o(o(oo)))

Robert Israel, Table of n, a(n) for n = 1..4000

The enumeration of these trees by nodes is A324969 (essentially A000045).
The enumeration of these trees by leaves appears to be A090129(n + 1).
The (non-semi) lone-child-avoiding version is A331683.
Matula-Goebel numbers of rooted semi-identity trees are A306202.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
The set S of numbers with at most one prime index in S is A331784.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Cf. A000081, A000669, A001678, A007097, A061775, A196050, A291636, A316470, A316473, A331679, A331680, A331682, A331687, A331783, A331785, A331873.

N:= 1000: # for terms <= N
S:= {1,2}:
with(queue):
Q:= new(1,2):
while not empty(Q) do
  r:= dequeue(Q);
  p:= ithprime(r);
  newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;
  S:= S union newS;
  for s in newS do enqueue(Q,s) od:
od:
sort(convert(S,list)); # Robert Israel, Feb 05 2020

uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,},{p,1}}/;uryQ[PrimePi[p]])|{{2,_}}];
Select[Range[100],uryQ]

Intersection of A306202 (semi-identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding). - Gus Wiseman, Jun 09 2020

A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.

Original entry on oeis.org

2, 3, 8, 24, 67, 214, 687, 2406, 8672, 32641, 125431, 493039, 1964611

Offset: 1

Gus Wiseman, Feb 02 2020

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.

			The a(1) = 2 through a(4) = 24 trees:
  o    (oo)      (ooo)          (oooo)
  (o)  (o(o))    (o(oo))        (o(ooo))
       ((o)(o))  (oo(o))        (oo(oo))
                 (o(o)(o))      (ooo(o))
                 (o(o(o)))      ((oo)(oo))
                 ((o)(o)(o))    (o(o(oo)))
                 (o((o)(o)))    (o(oo(o)))
                 ((o)((o)(o)))  (oo(o)(o))
                                (oo(o(o)))
                                (o(o)(o)(o))
                                (o(o(o)(o)))
                                (o(o(o(o))))
                                (oo((o)(o)))
                                ((o)(o)(o)(o))
                                ((o(o))(o(o)))
                                ((oo)((o)(o)))
                                (o((o)(o)(o)))
                                (o(o)((o)(o)))
                                (o(o((o)(o))))
                                ((o)((o)(o)(o)))
                                ((o)(o)((o)(o)))
                                (o((o)((o)(o))))
                                (((o)(o))((o)(o)))
                                ((o)((o)((o)(o))))

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.

Not requiring local disjointness gives A050381.
The non-semi version is A316697.
The same trees counted by number of vertices are A331872.
The Matula-Goebel numbers of these trees are A331873.
Lone-child-avoiding rooted trees counted  by leaves are A000669.
Semi-lone-child-avoiding rooted trees counted by vertices are A331934.
Cf. A000081, A001678, A300660, A316473, A316495, A316696, A331678, A331679, A331680, A331682, A331687, A331871, A331935.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
slaurt[n_]:=If[n==1,{o,{o}},Join@@Table[Select[Union[Sort/@Tuples[slaurt/@ptn]],disjointQ[Select[#,!AtomQ[#]&]]&],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[slaurt[n]],{n,8}]

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

Original entry on oeis.org

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

Gus Wiseman, Feb 02 2020

nonn

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.

			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}

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.

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.

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

Intersection of A291636 and A316495.

A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 37, 83, 191, 450, 1076, 2610, 6404, 15875, 39676, 99880, 253016, 644524, 1649918, 4242226

Offset: 1

Gus Wiseman, Jan 31 2020

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct.

			The a(1) = 1 through a(6) = 17 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o(o)))   (((ooo)))
                          (o((o)))   ((o(oo)))
                          ((((o))))  ((oo(o)))
                                     (o((oo)))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o(o))))
                                     ((o)((o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

The lone-child-avoiding case is A212804.
The identity tree version is A316471.
The Matula-Goebel numbers of these trees are given by A331682.
Identity trees are A004111.
Semi-identity trees are A306200.
Locally disjoint rooted trees are A316473.
Matula-Goebel numbers of locally disjoint semi-identity trees are A316494.
Cf. A000081, A306202, A316475, A316495, A316694, A331683, A331684, A331686.

disjunsQ[u_]:=Length[u]==1||UnsameQ@@DeleteCases[u,{}]&&Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldrsi[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[ldrsi/@c]]]/@IntegerPartitions[n-1],disjunsQ]];
Table[Length[ldrsi[n]],{n,10}]

cp's OEIS Frontend

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

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A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

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A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

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A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.

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

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A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

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