A109129 -id:A109129 - cp's OEIS frontend

A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

1, 4, 12, 14, 18, 19, 21, 27, 40, 52, 60, 68, 70, 74, 78, 86, 89, 90, 91, 92, 95, 100, 102, 105, 107, 111, 117, 119, 122, 129, 130, 134, 135, 138, 146, 150, 151, 153, 161, 163, 169, 170, 175, 176, 181, 183, 185, 195, 201, 206, 207, 215, 219, 221, 225, 227, 230

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

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.

			The terms together with their corresponding rooted trees begin:
   1: o
   4: (oo)
  12: (oo(o))
  14: (o(oo))
  18: (o(o)(o))
  19: ((ooo))
  21: ((o)(oo))
  27: ((o)(o)(o))
  40: (ooo((o)))
  52: (oo(o(o)))
  60: (oo(o)((o)))
  68: (oo((oo)))
  70: (o((o))(oo))
  74: (o(oo(o)))
  78: (o(o)(o(o)))
  86: (o(o(oo)))
  89: ((ooo(o)))
  90: (o(o)(o)((o)))

Gus Wiseman, The first 64 square rooted trees.

Internals instead of leaves: A358576, counted by A358587, ordered A358588.
Internals instead of height: A358578, counted by A185650, ordered A358579.
These trees are counted by A358589, ordered A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358379, A358580, A358581-A358586, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],{},{0,Infinity}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A109129(a(n)).

A358552 Node-height of the rooted tree with Matula-Goebel number n. Number of nodes in the longest path from root to leaf.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 2, 3, 4, 5, 3, 4, 3, 4, 2, 4, 3, 3, 4, 3, 5, 4, 3, 4, 4, 3, 3, 5, 4, 6, 2, 5, 4, 4, 3, 4, 3, 4, 4, 5, 3, 4, 5, 4, 4, 5, 3, 3, 4, 4, 4, 3, 3, 5, 3, 3, 5, 5, 4, 4, 6, 3, 2, 4, 5, 4, 4, 4, 4, 5, 3, 4, 4, 4, 3, 5, 4, 6, 4, 3, 5, 5, 3, 4, 4, 5, 5, 4, 4, 4, 4, 6, 5, 4, 3, 5, 3, 5, 4, 5, 4, 4, 4, 4, 3, 4, 3

Offset: 1

Gus Wiseman, Nov 26 2022

nonn

Edge-height is given by A109082 (see formula).

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.

			The Matula-Goebel number of ((ooo(o))) is 89, and it has node-height 4, so a(89) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of first appearances are A007097.
This statistic is counted by A034781, ordered A080936.
The ordered version is A358379(n) + 1.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
Other statistics: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A342507 (internals).
Cf. A000040, A000720, A001222, A056239, A112798.
Cf. A206487, A358576, A358589, A358592, A358729.

MGTree[n_]:=If[n==1,{},MGTree/@If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Depth[MGTree[n]]-1,{n,100}]

A358552(n) = { my(v=factor(n)[, 1], d=0); while(#v, d++; v=fold(setunion, apply(p->factor(primepi(p))[, 1]~, v))); (1+d); }; \\ (after Kevin Ryde in A109082) - Antti Karttunen, Oct 23 2023

from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
def A358552(n):
    if n == 1 : return 1
    if isprime(n): return 1+A358552(primepi(n))
    return max(A358552(p) for p in primefactors(n)) # Chai Wah Wu, Apr 15 2024

a(n) = A109082(n) + 1.

a(n) = A061775(n) - A358729(n). - Antti Karttunen, Oct 23 2023

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 28, 8, 36, 60, 330, 24, 156, 56, 180, 16, 476, 72, 152, 120, 168, 660, 828, 48, 900, 312, 216, 112, 1740, 360, 10230, 32, 1980, 952, 840, 144, 888, 304, 936, 240, 6396, 336, 2408, 1320, 1080, 1656, 8460, 96, 784, 1800, 2856, 624, 848, 432

Offset: 1

Gus Wiseman, Mar 20 2019

Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0 given by the rows of A324924. Then a(n) is the number obtained by encoding this factorization as a standard factorization into prime numbers (A112798).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Sorting the sequence gives A324850.
Cf. A000081, A003963, A061775, A109129, A112798, A120383, A196050, A317713.
Cf. A324848, A324849, A324923, A324924, A324925, A324931, A324934.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
difac[n_]:=If[n==1,{},With[{m=Product[Prime[i]/i,{i,primeMS[n]}]},Sort[Join[primeMS[n],difac[n/m]]]]];
Table[Times@@Prime/@difac[n],{n,30}]

a(n) = my (f=factor(n)); prod (i=1, #f~, (f[i,1] * a(primepi(f[i,1])))^f[i,2]) \\ Rémy Sigrist, Jul 18 2019

a(n) = Product_t mg(t) where the product is over all (not necessarily distinct) terminal subtrees of the rooted tree with Matula-Goebel number n, and mg(t) is the Matula-Goebel number of t.

Completely multiplicative with a(prime(n)) = prime(n) * a(n). - Rémy Sigrist, Jul 18 2019

Keyword mult added by Rémy Sigrist, Jul 18 2019

A358578 Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

Original entry on oeis.org

2, 6, 7, 18, 20, 21, 26, 34, 37, 43, 54, 60, 63, 67, 70, 78, 88, 91, 92, 95, 102, 111, 116, 119, 122, 129, 142, 146, 151, 162, 164, 173, 180, 181, 189, 200, 201, 202, 210, 227, 234, 236, 239, 245, 260, 264, 269, 273, 276, 278, 285, 306, 308, 314, 322, 333, 337

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

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.

			The terms together with their corresponding rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  26: (o(o(o)))
  34: (o((oo)))
  37: ((oo(o)))
  43: ((o(oo)))
  54: (o(o)(o)(o))
  60: (oo(o)((o)))
  63: ((o)(o)(oo))
  67: (((ooo)))
  70: (o((o))(oo))
  78: (o(o)(o(o)))
  88: (ooo(((o))))
  91: ((oo)(o(o)))

Gus Wiseman, The first 64 rooted trees whose number of leaves equals their number of internal nodes.

These trees are counted by A185650, ordered A358579.
Height instead of leaves: A358576, counted by A358587, ordered A358588.
Height instead of internals: A358577, counted by A358589, ordered A358590.
Positions of 0's in A358580.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358371, A358581-A358586, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],{},{0,Infinity}]==Count[MGTree[#],[_],{0,Infinity}]&]

A342507(a(n)) = A109129(a(n)).

A358589 Number of square rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 11, 17, 55, 107, 317, 720, 1938, 4803, 12707, 32311, 85168, 220879, 581112, 1522095, 4014186, 10568936, 27934075, 73826753, 195497427, 517927859, 1373858931, 3646158317, 9684878325, 25737819213, 68439951884, 182070121870, 484583900955, 1290213371950

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

We say that a tree is square if it has the same height as number of leaves.

			The a(1) = 1 through a(7) = 11 trees:
  o  .  (oo)  .  ((ooo))  ((o)(oo))  (((oooo)))
                 (o(oo))  (o(o)(o))  ((o(ooo)))
                 (oo(o))             ((oo(oo)))
                                     ((ooo(o)))
                                     (o((ooo)))
                                     (o(o(oo)))
                                     (o(oo(o)))
                                     (oo((oo)))
                                     (oo(o(o)))
                                     (ooo((o)))
                                     ((o)(o)(o))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For internals instead of height we have A185650 aerated, ranked by A358578.
These trees are ranked by A358577.
For internals instead of leaves we have A358587, ranked by A358576.
The ordered version is A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
Cf. A000891, A065097, A109129, A358552, A358588, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]==Depth[#]-1&]],{n,1,10}]

\\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1  + exp( sum(i=1, n-1, 1/i * subst( subst( A + O(x*x^((n-1)\i)), x, x^i), y, y^i) ) )); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

A358576 Matula-Goebel numbers of rooted trees whose node-height equals their number of internal (non-leaf) nodes.

Original entry on oeis.org

9, 15, 18, 21, 23, 30, 33, 35, 36, 39, 42, 46, 47, 49, 51, 57, 60, 61, 66, 70, 72, 73, 77, 78, 83, 84, 87, 91, 92, 93, 94, 95, 98, 102, 111, 113, 114, 119, 120, 122, 123, 129, 132, 133, 137, 140, 144, 146, 149, 151, 154, 156, 159, 166, 167, 168, 174, 177, 181

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

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.

Node-height is the number of nodes in the longest path from root to leaf.

			The terms together with their corresponding rooted trees begin:
   9: ((o)(o))
  15: ((o)((o)))
  18: (o(o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  30: (o(o)((o)))
  33: ((o)(((o))))
  35: (((o))(oo))
  36: (oo(o)(o))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  46: (o((o)(o)))
  47: (((o)((o))))
  49: ((oo)(oo))
  51: ((o)((oo)))
  57: ((o)(ooo))
  60: (oo(o)((o)))
  61: ((o(o)(o)))

Gus Wiseman, The first 64 rooted trees whose node-height equals their number of internal nodes.

The version for edge-height is A209638.
Square trees are A358577, counted by A358589, ordered A358590.
The version for leaves instead of height is A358578, counted by A185650.
These trees are counted by A358587, ordered A358588.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358580, A358581-A358586, A358592, A358729.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],[_],{0,Infinity}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A342507(a(n)).

A358580 Difference between the number of leaves and the number of internal (non-leaf) nodes in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 0, -1, 1, -2, 0, 0, 2, -1, -1, -3, 1, -1, 1, -2, 3, -1, 0, 1, 0, 0, -2, -2, 2, -3, 0, -1, 2, -2, -1, -4, 4, -3, 0, -1, 1, 0, 2, -1, 1, -2, 1, 0, -1, -2, -1, -3, 3, 1, -2, -1, 1, 2, 0, -4, 3, 1, -1, -2, 0, -1, -3, 0, 5, -2, -2, 0, 1, -2, 0, -1, 2, -1, 1, -3

Offset: 1

Gus Wiseman, Nov 25 2022

sign

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.

			The Matula-Goebel number of ((ooo(o))) is 89, and it has 4 leaves and 3 internal nodes, so a(89) = 1.

Zeros are A358578, counted by A185650 (ordered A358579).
Positions of positive terms are counted by A358581, negative A358582.
Positions of nonnegative terms are counted by A358583, nonpositive A358584.
A000081 counts rooted trees, ordered A000108.
A034781 counts trees by nodes and height.
A055277 counts trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358371, A358576, A358577, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],{},{0,Infinity}]-Count[MGTree[n],[_],{0,Infinity}],{n,100}]

a(n) = A109129(n) - A342507(n).

A358590 Number of square ordered rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 6, 5, 36, 84, 309, 890, 3163, 9835, 32979, 108252, 360696, 1192410, 3984552, 13276769, 44371368, 148402665, 497072593, 1665557619, 5586863093, 18750662066, 62968243731, 211565969511, 711187790166, 2391640404772, 8045964959333, 27077856222546

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

We say that a tree is square if it has the same height as number of leaves.

			The a(1) = 1 through a(6) = 5 ordered trees:
  o  .  (oo)  .  ((o)oo)  ((o)(o)o)
                 ((oo)o)  ((o)(oo))
                 ((ooo))  ((o)o(o))
                 (o(o)o)  ((oo)(o))
                 (o(oo))  (o(o)(o))
                 (oo(o))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For internals instead of height we have A000891, unordered A185650 aerated.
For internals instead of leaves we have A358588, unordered A358587.
The unordered version is A358589, ranked by A358577.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
Cf. A065097, A109129, A358371, A358552, A358579, A358586.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]==Depth[#]-1&]],{n,1,10}]

\\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1  + 1/(1 - A + O(x^n))); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(16) and beyond from Andrew Howroyd, Jan 01 2023

A358372 Number of nodes in the n-th standard ordered rooted tree.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 6, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 8, 8

Offset: 1

Gus Wiseman, Nov 14 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The standard ordered rooted tree ranking begins:
  1: o        10: (((o))o)   19: (((o))(o))
  2: (o)      11: ((o)(o))   20: (((o))oo)
  3: ((o))    12: ((o)oo)    21: ((o)((o)))
  4: (oo)     13: (o((o)))   22: ((o)(o)o)
  5: (((o)))  14: (o(o)o)    23: ((o)o(o))
  6: ((o)o)   15: (oo(o))    24: ((o)ooo)
  7: (o(o))   16: (oooo)     25: (o(oo))
  8: (ooo)    17: ((((o))))  26: (o((o))o)
  9: ((oo))   18: ((oo)o)    27: (o(o)(o))
For example, the 25th ordered tree is (o,(o,o)) because the 24th composition is (1,4) and the 3rd composition is (1,1). Hence a(25) = 5.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The triangle counting trees by leaves is A001263, unordered A055277.
The version for unordered trees is A061775, leaves A109129, edges A196050.
The leaves are counted by A358371.
A000081 counts unlabeled rooted trees, ranked by A358378.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120
Cf. A001678, A004249, A005043, A063895, A284778, A358373, A358376, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Table[Count[srt[n],_,{0,Infinity}],{n,100}]

A358586 Number of ordered rooted trees with n nodes, at least half of which are leaves.

Original entry on oeis.org

1, 1, 1, 4, 7, 31, 66, 302, 715, 3313, 8398, 39095, 104006, 484706, 1337220, 6227730, 17678835, 82204045, 238819350, 1108202513, 3282060210, 15195242478, 45741281820, 211271435479, 644952073662, 2971835602526, 9183676536076, 42217430993002, 131873975875180, 604834233372884

Offset: 1

Gus Wiseman, Nov 24 2022

nonn

			The a(1) = 1 through a(5) = 7 ordered trees:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                ((oo))  ((oo)o)
                (o(o))  ((ooo))
                        (o(o)o)
                        (o(oo))
                        (oo(o))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

For equality we have A000891, unordered A185650.
Odd-indexed terms appear to be A065097.
The unordered version is A358583.
The opposite is the same, unordered A358584.
The strict case is A358585, unordered A358581.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
A358590 counts square ordered trees, unordered A358589 (ranked by A358577).
Cf. A109129, A342507, A358371, A358579, A358580, A358582, A358584, A358588.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]>=Count[#,[_],{0,Infinity}]&]],{n,1,10}]

a(n) = if(n==1, 1, n--; (binomial(2*n,n)/(n+1) + if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ Andrew Howroyd, Jan 13 2024

From Andrew Howroyd, Jan 13 2024: (Start)
a(n) = Sum_{k=1..floor(n/2)} A001263(n-1, k) for n >= 2.
a(2*n) = (A000108(2*n-1) + A000891(n-1))/2 for n >= 1;
a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)

a(16) onwards from Andrew Howroyd, Jan 13 2024

cp's OEIS Frontend

A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

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A358552 Node-height of the rooted tree with Matula-Goebel number n. Number of nodes in the longest path from root to leaf.

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A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.

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A358578 Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

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A358589 Number of square rooted trees with n nodes.

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A358576 Matula-Goebel numbers of rooted trees whose node-height equals their number of internal (non-leaf) nodes.

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A358580 Difference between the number of leaves and the number of internal (non-leaf) nodes in the rooted tree with Matula-Goebel number n.

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