A331993 -id:A331993 - cp's OEIS frontend

A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 133, 152, 172, 212, 214, 224, 256, 262, 266, 301, 304, 326, 344, 371, 424, 428, 448, 512, 524, 526, 532, 602, 608, 622, 652, 688, 742, 749, 766, 817, 848, 856, 886, 896, 917, 1007, 1024, 1048, 1052

Offset: 1

Gus Wiseman, Feb 04 2020

nonn

First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)).
Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf 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.
Consists of one, and all composite numbers that are n times a power of two, where n is a squarefree number whose prime indices already belong to the sequence, and a prime index of n is a number m such that prime(m) divides n. [Clarified by Peter Munn and Gus Wiseman, Jun 24 2021]

			The sequence of all lone-child-avoiding rooted semi-identity 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))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
The sequence of terms together with their prime indices begins:
    1: {}                 224: {1,1,1,1,1,4}
    4: {1,1}              256: {1,1,1,1,1,1,1,1}
    8: {1,1,1}            262: {1,32}
   14: {1,4}              266: {1,4,8}
   16: {1,1,1,1}          301: {4,14}
   28: {1,1,4}            304: {1,1,1,1,8}
   32: {1,1,1,1,1}        326: {1,38}
   38: {1,8}              344: {1,1,1,14}
   56: {1,1,1,4}          371: {4,16}
   64: {1,1,1,1,1,1}      424: {1,1,1,16}
   76: {1,1,8}            428: {1,1,28}
   86: {1,14}             448: {1,1,1,1,1,1,4}
  106: {1,16}             512: {1,1,1,1,1,1,1,1,1}
  112: {1,1,1,1,4}        524: {1,1,32}
  128: {1,1,1,1,1,1,1}    526: {1,56}
  133: {4,8}              532: {1,1,4,8}
  152: {1,1,1,8}          602: {1,4,14}
  172: {1,1,14}           608: {1,1,1,1,1,8}
  212: {1,1,16}           622: {1,64}
  214: {1,28}             652: {1,1,38}

The non-semi case is {1}.
Not requiring lone-child-avoidance gives A306202.
The locally disjoint version is A331683.
These trees are counted by A331966.
The semi-lone-child-avoiding case is A331994.
Matula-Goebel numbers of rooted identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Semi-identity trees are counted by A306200.
Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.

csiQ[n_]:=n==1||!PrimeQ[n]&&FreeQ[FactorInteger[n],{?(#>2&),?(#>1&)}]&&And@@csiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],csiQ]

Intersection of A291636 and A306202.

A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 55, 105, 200, 388, 754, 1483, 2923, 5807, 11575, 23190, 46608, 94043, 190287, 386214, 785831, 1602952, 3276845, 6712905, 13778079, 28330583, 58350582, 120370731, 248676129, 514459237, 1065696295, 2210302177, 4589599429, 9540623926

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

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

In a semi-identity tree, the non-leaf branches of any given vertex are distinct.

			The a(1) = 1 through a(9) = 16 trees (empty column shown as dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        (o(o(oo)))  (oooo(oo))   (oooo(ooo))
                                                    ((oo)(ooo))  (ooooo(oo))
                                                    (o(o(ooo)))  ((oo)(oooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo)(ooo))
                                                                 (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 ((oo)(o(oo)))
                                                                 (o(o(o(oo))))

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The non-semi case is A000007.
Lone-child-avoiding rooted trees are A001678.
The locally disjoint case is A212804.
Not requiring lone-child-avoidance gives A306200.
Matula-Goebel numbers of these trees are A331965.
The semi-lone-child-avoiding version is A331993.
Cf. A000081, A004111, A291636, A300660, A306202, A316694, A331683, A331686, A331783, A331875, A331964, A331994.

ssb[n_]:=If[n==1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[ssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssb[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[0, 0]); for(n=2, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020

Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020

A331994 Matula-Goebel numbers of semi-lone-child-avoiding rooted semi-identity trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 21, 24, 26, 28, 32, 38, 39, 42, 48, 52, 56, 57, 64, 74, 76, 78, 84, 86, 91, 96, 104, 106, 111, 112, 114, 128, 129, 133, 146, 148, 152, 156, 159, 168, 172, 178, 182, 192, 202, 208, 212, 214, 219, 222, 224, 228, 247, 256, 258, 259, 262

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.
In a semi-identity tree, the non-leaf branches of any given vertex are 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.
Consists of one, two, and all numbers that can be written as a power of two (other than 2) times a squarefree number whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.

			The sequence of all semi-lone-child-avoiding rooted semi-identity trees 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)
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  48: (oooo(o))
  52: (oo(o(o)))
  56: (ooo(oo))
  57: ((o)(ooo))
The sequence of terms together with their prime indices begins:
    1: {}              64: {1,1,1,1,1,1}      159: {2,16}
    2: {1}             74: {1,12}             168: {1,1,1,2,4}
    4: {1,1}           76: {1,1,8}            172: {1,1,14}
    6: {1,2}           78: {1,2,6}            178: {1,24}
    8: {1,1,1}         84: {1,1,2,4}          182: {1,4,6}
   12: {1,1,2}         86: {1,14}             192: {1,1,1,1,1,1,2}
   14: {1,4}           91: {4,6}              202: {1,26}
   16: {1,1,1,1}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   21: {2,4}          104: {1,1,1,6}          212: {1,1,16}
   24: {1,1,1,2}      106: {1,16}             214: {1,28}
   26: {1,6}          111: {2,12}             219: {2,21}
   28: {1,1,4}        112: {1,1,1,1,4}        222: {1,2,12}
   32: {1,1,1,1,1}    114: {1,2,8}            224: {1,1,1,1,1,4}
   38: {1,8}          128: {1,1,1,1,1,1,1}    228: {1,1,2,8}
   39: {2,6}          129: {2,14}             247: {6,8}
   42: {1,2,4}        133: {4,8}              256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    146: {1,21}             258: {1,2,14}
   52: {1,1,6}        148: {1,1,12}           259: {4,12}
   56: {1,1,1,4}      152: {1,1,1,8}          262: {1,32}
   57: {2,8}          156: {1,1,2,6}          266: {1,4,8}

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The locally disjoint version is A331681.
The enumeration of these trees by vertices is A331993.
Semi-identity trees are A306200.
MG-numbers of rooted identity trees are A276625.
MG-numbers of lone-child-avoiding rooted identity trees are {1}.
MG-numbers of lone-child-avoiding rooted trees are A291636.
MG-numbers of semi-identity trees are A306202.
MG-numbers of semi-lone-child-avoiding rooted trees are A331935.
MG-numbers of semi-lone-child-avoiding rooted identity trees are A331963.
MG-numbers of lone-child-avoiding rooted semi-identity trees are A331965.
Cf. A004111, A007097, A061775, A122132, A196050, A320269, A331683, A331873, A331934, A331936, A331964, A331966.

scsiQ[n_]:=n==1||n==2||!PrimeQ[n]&&FreeQ[FactorInteger[n],{?(#>2&),?(#>1&)}]&&And@@scsiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],scsiQ]

Intersection of A306202 and A331935.

A331991 Number of semi-lone-child-avoiding achiral rooted trees with n vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 4, 4, 1, 7, 1, 7, 5, 6, 1, 7, 3, 7, 5, 7, 1, 13, 1, 8, 6, 6, 6, 10, 1, 9, 7, 9, 1, 15, 1, 12, 12, 8, 1, 12, 4, 13, 6, 11, 1, 15, 7, 13, 9, 9, 1, 17, 1, 15, 15, 9, 8, 21, 1, 13, 8, 16, 1, 18, 1, 12, 16, 11, 8, 21, 1

Offset: 1

Gus Wiseman, Feb 06 2020

nonn

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

In an achiral rooted tree, the branches of any given vertex are all equal.

			The a(n) trees for n = 2, 3, 5, 7, 11, 13:
  (o)  (oo)  (oooo)    (oooooo)     (oooooooooo)        (oooooooooooo)
             ((o)(o))  ((oo)(oo))   ((oooo)(oooo))      ((ooooo)(ooooo))
                       ((o)(o)(o))  ((o)(o)(o)(o)(o))   ((ooo)(ooo)(ooo))
                                    (((o)(o))((o)(o)))  ((oo)(oo)(oo)(oo))
                                                        ((o)(o)(o)(o)(o)(o))

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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.

Matula-Goebel numbers of these trees are A331992.
The fully lone-child-avoiding case is A167865.
The semi-achiral version is A331933.
Not requiring achirality gives A331934.
The identity tree version is A331964.
The semi-identity tree version is A331993.
Achiral rooted trees are counted by A003238.
Lone-child-avoiding semi-achiral trees are A320268.
Cf. A000081, A004111, A050381, A001678, A214577, A289079, A320222, A331912, A331935, A331936, A331963, A331967, A331994.

ab[n_]:=If[n<=2,1,Sum[ab[d],{d,Most[Divisors[n-1]]}]];
Array[ab,100]

a(1) = a(2) = 1; a(n + 1) = Sum_{d|n, d 1.

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 + x)) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, Feb 25 2020

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A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

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A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

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A331994 Matula-Goebel numbers of semi-lone-child-avoiding rooted semi-identity trees.

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A331991 Number of semi-lone-child-avoiding achiral rooted trees with n vertices.

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