A331992 -id:A331992 - cp's OEIS frontend

A167865 Number of partitions of n into distinct parts greater than 1, with each part divisible by the next.

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 3, 3, 3, 1, 5, 1, 5, 4, 3, 1, 6, 2, 5, 4, 5, 1, 9, 1, 6, 4, 4, 4, 8, 1, 6, 6, 7, 1, 11, 1, 8, 8, 4, 1, 10, 3, 10, 5, 8, 1, 11, 4, 10, 7, 6, 1, 13, 1, 10, 11, 7, 6, 15, 1, 9, 5, 11, 1, 14, 1, 9, 12, 8, 5, 15, 1, 16, 9, 8, 1, 18, 5, 12, 7, 10, 1, 21, 7, 13, 11, 5

Offset: 0

Max Alekseyev, Nov 13 2009

Number of lone-child-avoiding achiral rooted trees with n + 1 vertices, where a rooted tree is lone-child-avoiding if all terminal subtrees have at least two branches, and achiral if all branches directly under any given vertex are equal. The Matula-Goebel numbers of these trees are given by A331967. - Gus Wiseman, Feb 07 2020

			a(12) = 4: [12], [10,2], [9,3], [8,4].
a(14) = 3: [14], [12,2], [8,4,2].
a(18) = 5: [18], [16,2], [15,3], [12,6], [12,4,2].
From _Gus Wiseman_, Jul 13 2018: (Start)
The a(36) = 8 lone-child-avoiding achiral rooted trees with 37 vertices:
  (oooooooooooooooooooooooooooooooooooo)
  ((oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo))
  ((ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo))
  ((ooooo)(ooooo)(ooooo)(ooooo)(ooooo)(ooooo))
  ((oooooooo)(oooooooo)(oooooooo)(oooooooo))
  (((ooo)(ooo))((ooo)(ooo))((ooo)(ooo))((ooo)(ooo)))
  ((ooooooooooo)(ooooooooooo)(ooooooooooo))
  ((ooooooooooooooooo)(ooooooooooooooooo))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A001678, A067824, A122651, A167439, A167865, A167866, A184998, A316782.
The semi-achiral version is A320268.
Matula-Goebel numbers of these trees are A331967.
The semi-lone-child-avoiding version is A331991.
Achiral rooted trees are counted by A003238.
Cf. A000081, A000669, A289079, A320222, A331912, A331933, A331936, A331992.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(a((n-d)/d), d=divisors(n) minus{1}))
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Mar 28 2011

a[0] = 1; a[n_] := a[n] = DivisorSum[n, a[(n-#)/#]&, #>1&]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 07 2015 *)

{ A167865(n) = if(n==0,return(1)); sumdiv(n,d, if(d>1, A167865((n-d)\d) ) ) }

a(0) = 1 and for n>=1, a(n) = Sum_{d|n, d>1} a((n-d)/d).

G.f. A(x) satisfies: A(x) = 1 + x^2*A(x^2) + x^3*A(x^3) + x^4*A(x^4) + ... - Ilya Gutkovskiy, May 09 2019

A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

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, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

First differs from A331873 in lacking 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 that child is an endpoint/leaf.
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 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.

			The sequence of rooted trees ranked by this sequence 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))
The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}            152: {1,1,1,8}
    2: {1}             54: {1,2,2,2}          162: {1,2,2,2,2}
    4: {1,1}           56: {1,1,1,4}          169: {6,6}
    6: {1,2}           64: {1,1,1,1,1,1}      172: {1,1,14}
    8: {1,1,1}         72: {1,1,1,2,2}        178: {1,24}
    9: {2,2}           74: {1,12}             184: {1,1,1,9}
   12: {1,1,2}         76: {1,1,8}            192: {1,1,1,1,1,1,2}
   14: {1,4}           81: {2,2,2,2}          196: {1,1,4,4}
   16: {1,1,1,1}       86: {1,14}             202: {1,26}
   18: {1,2,2}         92: {1,1,9}            206: {1,27}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   26: {1,6}           98: {1,4,4}            212: {1,1,16}
   27: {2,2,2}        104: {1,1,1,6}          214: {1,28}
   28: {1,1,4}        106: {1,16}             216: {1,1,1,2,2,2}
   32: {1,1,1,1,1}    108: {1,1,2,2,2}        224: {1,1,1,1,1,4}
   36: {1,1,2,2}      112: {1,1,1,1,4}        243: {2,2,2,2,2}
   38: {1,8}          122: {1,18}             244: {1,1,18}
   46: {1,9}          128: {1,1,1,1,1,1,1}    256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    144: {1,1,1,1,2,2}      262: {1,32}
   49: {4,4}          148: {1,1,12}           288: {1,1,1,1,1,2,2}

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.

A superset of A000079.
The non-lone-child-avoiding version is A320230.
The non-semi version is A320269.
These trees are counted by A331933.
Not requiring semi-achirality gives A331935.
The fully-achiral case is A331992.
Achiral trees are counted by A003238.
Numbers with at most one distinct odd prime factor are A070776.
Matula-Goebel numbers of achiral rooted trees are A214577.
Matula-Goebel numbers of semi-identity trees are A306202.
Numbers S with at most one distinct prime index in S are A331912.
Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n],{2,_}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

Intersection of A320230 and A331935.

A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 823543, 1048576, 2097152, 2248091, 2476099, 2621161, 4194304

Offset: 1

Gus Wiseman, Feb 06 2020

nonn

Lone-child-avoiding means there are no unary branchings.
In an achiral rooted tree, the branches of any given vertex are all equal.
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 numbers of the form prime(j)^k where k > 1 and j is already in the sequence.

			The sequence of all lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
      1: o
      4: (oo)
      8: (ooo)
     16: (oooo)
     32: (ooooo)
     49: ((oo)(oo))
     64: (oooooo)
    128: (ooooooo)
    256: (oooooooo)
    343: ((oo)(oo)(oo))
    361: ((ooo)(ooo))
    512: (ooooooooo)
   1024: (oooooooooo)
   2048: (ooooooooooo)
   2401: ((oo)(oo)(oo)(oo))
   2809: ((oooo)(oooo))
   4096: (oooooooooooo)
   6859: ((ooo)(ooo)(ooo))
   8192: (ooooooooooooo)
  16384: (oooooooooooooo)
  16807: ((oo)(oo)(oo)(oo)(oo))
  17161: ((ooooo)(ooooo))
  32768: (ooooooooooooooo)
  51529: (((oo)(oo))((oo)(oo)))
  65536: (oooooooooooooooo)
  96721: ((oooooo)(oooooo))

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Gus Wiseman, The first 30 lone-child-avoiding achiral rooted trees.

A subset of A025475 (nonprime prime powers).
The enumeration of these trees by vertices is A167865.
Not requiring lone-child-avoidance gives A214577.
The semi-achiral version is A320269.
The semi-lone-child-avoiding version is A331992.
Achiral rooted trees are counted by A003238.
MG-numbers of planted achiral rooted trees are A280996.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A001678, A007097, A061775, A196050, A276625, A291441, A320230, A320268, A331912, A331936, A331963, A331965, A331966, A331991.

msQ[n_]:=n==1||!PrimeQ[n]&&PrimePowerQ[n]&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000],msQ]

Intersection of A214577 (achiral) and A291636 (lone-child-avoiding).

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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A167865 Number of partitions of n into distinct parts greater than 1, with each part divisible by the next.

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A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

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A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

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

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