A109129 -id:A109129 - cp's OEIS frontend

A324931 Integers in the list of quotients of positive integers by their product of prime indices.

1, 2, 4, 3, 8, 6, 16, 12, 7, 5, 32, 9, 24, 14, 10, 64, 18, 48, 28, 20, 128, 36, 19, 13, 21, 15, 96, 27, 56, 40, 256, 72, 38, 26, 11, 42, 30, 192, 54, 112, 17, 80, 512, 144, 76, 52, 22, 84, 60, 384, 49, 23, 35, 53, 108, 37, 224, 25, 57, 39, 34, 160, 63, 1024

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

These quotients are given by A324932(n)/A324933(n).

This is a permutation of the positive integers, with inverse A324934.

			The sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324934.

Select[Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}],IntegerQ]

a(n) = A324850(n)/A003963(A324850(n)).

A358373 Triangle read by rows where row n lists the sorted standard ordered rooted tree-numbers of all unlabeled ordered rooted trees with n vertices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 33, 65, 129, 257, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 41, 49, 50, 57, 66, 97, 130, 193, 258, 385, 513, 514, 769, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073

Offset: 1

Gus Wiseman, Nov 14 2022

We define the standard ordered rooted tree (SORT)-number of an unlabeled ordered rooted tree to be one plus the standard composition number (A066099) of the SORT-numbers of the branches, or 1 if there are no branches. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			Triangle begins:
   1
   2
   3   4
   5   6   7   8   9
  10  11  12  13  14  15  16  17  18  25  33  65 129 257
For example, the tree t = ((o,o),o) has branches (o,o) and o with SORT-numbers 4 and 1, and the standard composition number of (4,1) is 17, so t has SORT-number 18 and is found in row 5.

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

The version for compositions is A000027.
Row-lengths are A000108.
The unordered version (using Matula-Goebel numbers) is A061773.
The version for Heinz numbers of partitions is A215366.
The row containing n is A358372(n).
A000081 counts unlabeled rooted trees, ranked by A358378.
A001263 counts unlabeled ordered rooted trees by nodes and leaves.
A358371 counts leaves in standard ordered rooted trees.
Cf. A004249, A005043, A061775, A109129, A284778, A358376, A358377.

stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
aotrank[t_]:=If[t=={},1,1+stcinv[aotrank/@t]];
aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Sort[aotrank/@aot[n]],{n,6}]

A358581 Number of rooted trees with n nodes, most of which are leaves.

Original entry on oeis.org

1, 0, 1, 1, 4, 5, 20, 28, 110, 169, 663, 1078, 4217, 7169, 27979, 49191, 191440, 345771, 1341974, 2477719, 9589567, 18034670, 69612556, 132984290, 511987473, 991391707, 3807503552, 7460270591, 28585315026, 56595367747, 216381073935, 432396092153

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(1) = 1 through a(7) = 20 trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)
                     ((ooo))  ((oooo))  ((ooooo))
                     (o(oo))  (o(ooo))  (o(oooo))
                     (oo(o))  (oo(oo))  (oo(ooo))
                              (ooo(o))  (ooo(oo))
                                        (oooo(o))
                                        (((oooo)))
                                        ((o)(ooo))
                                        ((o(ooo)))
                                        ((oo)(oo))
                                        ((oo(oo)))
                                        ((ooo(o)))
                                        (o((ooo)))
                                        (o(o)(oo))
                                        (o(o(oo)))
                                        (o(oo(o)))
                                        (oo((oo)))
                                        (oo(o)(o))
                                        (oo(o(o)))
                                        (ooo((o)))

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

For equality we have A185650 aerated, ranked by A358578.
The opposite version is A358582, non-strict A358584.
The non-strict version is A358583.
The ordered version is A358585, odd-indexed terms A065097.
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.
A358589 counts square trees, ranked by A358577, ordered A358590.
Cf. A000891, A109129, A342507, A358579, A358580, A358586, 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}]>Count[#,[_],{0,Infinity}]&]],{n,0,10}]

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[n\2+1..#u]))} \\ Andrew Howroyd, Dec 31 2022

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=floor(n/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Terms a(19) and beyond from Andrew Howroyd, Dec 31 2022

A358379 Edge-height (or depth) of the n-th standard ordered rooted tree.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 1, 4, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 1, 3, 4, 2, 2, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 1, 3, 3, 4, 4, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 2, 3, 3, 3, 2, 2

Offset: 1

Gus Wiseman, Nov 16 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 52nd ordered tree is (o((o))oo) so a(52) = 3.

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

Records occur at A004249.
The triangle counting trees by this statistic is A080936, unordered A034781.
Unordered version is A109082, nodes A061775, leaves A109129, edges A196050.
Leaves are counted by A358371.
Nodes are counted by A358372.
Node-height is a(n) + 1, unordered version is A358552.
A000081 counts unordered rooted trees, ranked by A358378.
A000108 counts ordered rooted trees.
A001263 counts ordered rooted trees by leaves.
Cf. A005043, A055277, A187306, A358373, A358374, A358375, 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[Depth[srt[n]]-2,{n,100}]

A324925 Number of integer partitions y of n such that Product_{i in y} prime(i)/i is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 5, 8, 9, 11, 17, 19, 21, 28, 32, 40, 51, 57, 67, 83, 96, 118, 142, 160, 189, 224, 260, 307, 363, 412, 479, 561, 649, 749, 874, 997, 1141, 1321, 1518, 1734, 1994, 2274, 2582, 2960, 3374, 3837, 4370, 4950, 5604, 6371, 7208, 8157, 9231, 10392

Offset: 0

Gus Wiseman, Mar 20 2019

nonn

The Heinz numbers of these integer partitions are given by A324850.

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (11)  (21)   (211)   (2111)   (321)     (3211)     (32111)
             (111)  (1111)  (11111)  (411)     (4111)     (41111)
                                     (2211)    (22111)    (221111)
                                     (21111)   (211111)   (2111111)
                                     (111111)  (1111111)  (11111111)
For example, (3,2,1,1) is such a partition because (2/1) * (2/1) * (3/2) * (5/3) = 10 is an integer.

Cf. A003963, A109129, A324850, A324922, A324923, A324924, A324931, A324934.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Product[Prime[i]/i,{i,#}]]&]],{n,0,30}]

A358575 Triangle read by rows where T(n,k) is the number of unlabeled n-node rooted trees with k = 0..n-1 internal (non-leaf) nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 6, 1, 0, 1, 5, 14, 18, 9, 1, 0, 1, 6, 21, 39, 35, 12, 1, 0, 1, 7, 30, 72, 97, 62, 16, 1, 0, 1, 8, 40, 120, 214, 212, 103, 20, 1, 0, 1, 9, 52, 185, 416, 563, 429, 161, 25, 1

Offset: 1

Gus Wiseman, Nov 23 2022

			Triangle begins:
    1
    0    1
    0    1    1
    0    1    2    1
    0    1    3    4    1
    0    1    4    8    6    1
    0    1    5   14   18    9    1
    0    1    6   21   39   35   12    1
    0    1    7   30   72   97   62   16    1
    0    1    8   40  120  214  212  103   20    1
    0    1    9   52  185  416  563  429  161   25    1

Row sums are A000081.
Column k = n - 2 appears to be A002620.
Column k = 3 appears to be A006578.
The version for height instead of internal nodes is A034781.
Equals A055277 with rows reversed.
The ordered version is A090181 or A001263.
The central column is A185650, ordered A000891.
The left half sums to A358583, strict A358581.
The right half sums to A358584, strict A358582.
Cf. A000108, A065097, A109129, A342507, A358578, A358587, A358589.

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}]==k&]],{n,1,10},{k,0,n-1}]

A358592 Matula-Goebel numbers of rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

Original entry on oeis.org

18, 21, 60, 70, 78, 91, 92, 95, 102, 111, 119, 122, 129, 146, 151, 181, 201, 227, 264, 269, 308, 348, 376, 406, 418, 426, 452, 492, 497, 519, 551, 562, 574, 583, 596, 606, 659, 664, 668, 698, 707, 708, 717, 779, 794, 796, 809, 826, 834, 911, 932, 934, 942, 958

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:
   18: (o(o)(o))
   21: ((o)(oo))
   60: (oo(o)((o)))
   70: (o((o))(oo))
   78: (o(o)(o(o)))
   91: ((oo)(o(o)))
   92: (oo((o)(o)))
   95: (((o))(ooo))
  102: (o(o)((oo)))
  111: ((o)(oo(o)))
  119: ((oo)((oo)))
  122: (o(o(o)(o)))
  129: ((o)(o(oo)))
  146: (o((o)(oo)))
  151: ((oo(o)(o)))
  181: ((o(o)(oo)))
  201: ((o)((ooo)))
  227: (((oo)(oo)))

Gus Wiseman, The first 64 ordered trees whose height, number of leaves, and number of internal nodes are all equal.

Any number of leaves: A358576, counted by A358587 (ordered A358588).
Any number of internals: A358577, counted by A358589, ordered A358590.
Any height: A358578, ordered A358579, counted by A185650.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A206487, A358580, A358581-A358586.

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}]==Depth[MGTree[#]]-1&]

A358552(a(n)) = A342507(a(n)) = A109129(a(n)).

A291442 Matula-Goebel numbers of leaf-balanced trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 36, 37, 40, 41, 44, 45, 47, 48, 49, 50, 53, 54, 55, 59, 60, 61, 62, 64, 66, 67, 71, 72, 75, 79, 80, 81, 83, 88, 89, 90, 91, 93, 96, 97, 99, 100, 103, 108

Offset: 1

Gus Wiseman, Aug 23 2017

nonn

An unlabeled rooted tree is leaf-balanced if every branch has the same number of leaves and every non-leaf rooted subtree is also leaf-balanced.

Cf. A000081, A007097, A061775, A109129, A214577, A280994, A291441, A291443.

nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
balQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[SameQ@@leafcount/@m,And@@balQ/@m]]];
Select[Range[nn],balQ]

A291441 Matula-Goebel numbers of orderless same-trees with all leaves equal to 1.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 12031, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 637643, 718099, 757907, 823543, 1048576, 2097152, 2248091

Offset: 1

Gus Wiseman, Aug 23 2017

nonn

See A289078 for the definition of orderless same-tree.

			a(20)=12031 corresponds to the following same-tree: {{1,1,1,1},{{1,1},{1,1}}}.

Cf. A007097, A109129, A214577, A280994, A289078, A289079, A291442, A291443.

nn=200000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
sameQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]>1,SameQ@@leafcount/@m,And@@sameQ/@m]]];
Select[Range[nn],sameQ]

More terms from Jinyuan Wang, Jun 21 2020

A358729 Difference between the number of nodes and the node-height of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 1, 3, 2, 2, 3, 1, 2, 3, 3, 2, 4, 3, 1, 3, 0, 4, 2, 2, 3, 4, 2, 3, 3, 3, 1, 4, 2, 2, 4, 3, 2, 4, 4, 4, 3, 3, 3, 5, 3, 4, 4, 2, 1, 4, 3, 1, 5, 5, 4, 3, 2, 3, 4, 4, 2, 5, 3, 3, 5, 4, 3, 4, 1, 4, 6, 2, 2, 5, 4, 3, 3, 3, 3, 5, 4, 4, 2, 3, 4, 5, 3, 5, 4, 5, 2, 4, 4, 4, 5, 4, 3, 6

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Node-height is the number of nodes in the longest path from root to 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.
Because the number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n, i.e., A317713(n) (= 1+A324923(n)), is always at least one larger than the depth of the same tree (= A109082(n)), it follows that a(n) >= A366386(n) for all n. - Antti Karttunen, Oct 23 2023

			The tree (oo(oo(o))) with Matula-Goebel number 148 has 8 nodes and node-height 4, so a(148) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to Matula-Goebel numbers

Positions of 0's are A007097.
Positions of first appearances are A358730.
Positions of 1's are A358731.
Other differences: A358580, A358724, A358726.
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.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A056239, A112798.
Cf. A206487, A209638, A316321, A358576, A358577, A358592, A358725, A366386.

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

A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs,t,i); pfs=factor(n); pfs[,1]=apply(t->A061775(t),pfs[,1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i,1]*pfs[i,2])}));
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 program given in A109082 by Kevin Ryde, Sep 21 2020)
A358729(n) = (A061775(n)-A358552(n)); \\ Antti Karttunen, Oct 23 2023

a(n) = A061775(n) - A358552(n).

a(n) = A196050(n) - A109082(n). - Antti Karttunen, Oct 23 2023

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

cp's OEIS Frontend

A324931 Integers in the list of quotients of positive integers by their product of prime indices.

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A358373 Triangle read by rows where row n lists the sorted standard ordered rooted tree-numbers of all unlabeled ordered rooted trees with n vertices.

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A358581 Number of rooted trees with n nodes, most of which are leaves.

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A358379 Edge-height (or depth) of the n-th standard ordered rooted tree.

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A324925 Number of integer partitions y of n such that Product_{i in y} prime(i)/i is an integer.

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A358575 Triangle read by rows where T(n,k) is the number of unlabeled n-node rooted trees with k = 0..n-1 internal (non-leaf) nodes.

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A358592 Matula-Goebel numbers of rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

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A291442 Matula-Goebel numbers of leaf-balanced trees.

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A291441 Matula-Goebel numbers of orderless same-trees with all leaves equal to 1.