A214577 -id:A214577 - cp's OEIS frontend

A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.

1, 3, 5, 12, 17, 41, 65, 144, 262, 533, 1025, 2110, 4097, 8261, 16407, 32928, 65537, 131384, 262145, 524854, 1048647, 2098181, 4194305, 8390924, 16777234, 33558533, 67109132, 134226070, 268435457, 536887919, 1073741825, 2147516736, 4294968327, 8590000133

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

In these trees, achiral means that all branches directly under any given node that is not a leaf or a cover of leaves are equal, and series-reduced means that every node that is not a leaf or a cover of leaves has at least two branches.

			The a(4) = 12 trees:
  (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
  (1222),
  (1122), ((12)(12)),
  (1112),
  (1233),
  (1223),
  (1123),
  (1234).

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

Cf. A001678, A003238, A052409, A052410, A067824, A167865, A168532, A214577, A289078, A294336, A316782, A317099.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
b[n_]:=1+Sum[b[n/d],{d,Rest[Divisors[n]]}];
a[n_]:=Sum[b[GCD@@Length/@Split[ptn]],{ptn,allnorm[n]}];
Array[a,10]

seq(n)={my(v=vector(n)); for(n=1, n, v[n]=2^(n-1) + sumdiv(n, d, v[d])); v} \\ Andrew Howroyd, Aug 19 2018

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 07 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018

A317719 Numbers that are not powerful tree numbers.

Original entry on oeis.org

6, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 26, 28, 29, 30, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a powerful tree number iff either n = 1 or n is a prime number whose prime index is a powerful tree number, or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all powerful tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of numbers that are not powerful tree numbers together with their Matula-Goebel trees begins:
   6: (o(o))
  10: (o((o)))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Complement of A318612.
Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.
Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718.

powgoQ[n_]:=Or[n==1,If[PrimeQ[n],powgoQ[PrimePi[n]],And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!powgoQ[#]&]

A318993 Matula-Goebel number of the planted achiral tree determined by the n-th number whose consecutive prime indices are divisible.

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 16, 5, 9, 19, 32, 17, 64, 53, 11, 128, 23, 256, 67, 49, 131, 512, 59, 27, 311, 25, 241, 1024, 2048, 31, 719, 83, 4096, 1619, 361, 331, 8192, 227, 16384, 739, 3671, 32768, 277, 81, 103, 2063, 65536, 97, 1523, 2809, 8161, 131072, 262144, 17863

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of all planted achiral trees begins: o, (o), (oo), ((o)), (ooo), ((oo)), (oooo), (((o))), ((o)(o)), ((ooo)), (ooooo), (((oo))), (oooooo), ((oooo)), ((((o)))), (ooooooo), (((o)(o))), (oooooooo), (((ooo))), ((oo)(oo)).

Gus Wiseman, The first 81 planted achiral trees corresponding to the first 81 dividing partitions.

Cf. A000040, A001221, A003238, A008480, A061775, A214577, A318990, A318991, A318992.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ptnToAch[y_]:=Fold[Table[#1,{#2}]&,{},Divide@@@Partition[Append[y,1],2,1]];
MGNumber[[]]:=1;MGNumber[x:[__]]:=If[Length[x]==1,Prime[MGNumber[x[[1]]]],Times@@Prime/@MGNumber/@x];
MGNumber/@ptnToAch/@Reverse/@primeMS/@Select[Range[100],Or[#==1,PrimeQ[#],Divisible@@Reverse[primeMS[#]]]&]

A358507 Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).

Original entry on oeis.org

1, 6, 12, 24, 30, 48, 60, 72, 104, 120, 144, 148, 156, 180, 192, 222, 288, 312, 360, 390, 432, 444, 480, 576, 712, 720, 780, 832, 864, 900, 1080, 1110, 1248, 1260, 1296, 1440, 1560, 1680, 2136, 2160, 2262, 2304, 2340, 2496, 2520, 2592, 2738, 2880, 2886, 3072

Offset: 1

Gus Wiseman, Nov 20 2022

nonn

To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.

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 trees begin:
    1: o
    6: (o(o))
   12: (oo(o))
   24: (ooo(o))
   30: (o(o)((o)))
   48: (oooo(o))
   60: (oo(o)((o)))
   72: (ooo(o)(o))
  104: (ooo(o(o)))
  120: (ooo(o)((o)))
  144: (oooo(o)(o))
  148: (oo(oo(o)))
  156: (oo(o)(o(o)))
  180: (oo(o)(o)((o)))
  192: (oooooo(o))
  222: (o(o)(oo(o)))
  288: (ooooo(o)(o))
  312: (ooo(o)(o(o)))

Positions of first appearances in A206487.
The unsorted version is A358508.
A000081 counts rooted trees, ordered A000108.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A032027, A061775, A127301, A196050, A358378, A358506, A358521, A358522.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
treeperms[t_]:=Times@@Cases[t,b:{}:>Length[Permutations[b]],{0,Infinity}];
fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
fir[Table[treeperms[MGTree[n]],{n,100}]]

A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 22, 24, 32, 33, 34, 35, 36, 37, 38, 40, 43, 44, 48, 64, 66, 67, 68, 69, 70, 72, 74, 75, 76, 80, 86, 88, 96, 128, 129, 131, 132, 133, 134, 136, 137, 138, 139, 140, 144, 147, 148, 150, 152, 160, 171, 172

Offset: 1

Gus Wiseman, Nov 20 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.

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 terms together with their standard ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: ((o)o)
   8: (ooo)
   9: ((oo))
  10: (((o))o)
  11: ((o)(o))
  12: ((o)oo)
  16: (oooo)
  17: ((((o))))
  18: ((oo)o)
  19: (((o))(o))
  20: (((o))oo)

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

Positions of first appearances in A358506.
The unsorted version is A358522.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
fir[Table[mgnum[srt[n]],{n,100}]]

A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 11, 10, 17, 12, 33, 18, 19, 16, 257, 22, 129, 20, 35, 34, 1025, 24, 37, 66, 43, 36, 513, 38, 65537, 32, 67, 514, 69, 44, 2049, 258, 131, 40

Offset: 1

Gus Wiseman, Nov 20 2022

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 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 standard ordered trees begin:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    6: ((o)o)
    9: ((oo))
    8: (ooo)
   11: ((o)(o))
   10: (((o))o)
   17: ((((o))))
   12: ((o)oo)
   33: (((o)o))
   18: ((oo)o)
   19: (((o))(o))
   16: (oooo)
  257: (((oo)))
   22: ((o)(o)o)
  129: ((ooo))
   20: (((o))oo)
   35: ((oo)(o))
   34: ((((o)))o)

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

Position of first appearance of n in A358506.
The sorted version is A358521.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
uv=Table[mgnum[srt[n]],{n,10000}];
Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]

A298538 Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of nodes.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 187

Offset: 1

Gus Wiseman, Jan 21 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
4  (oo)
5  (((o)))
7  ((oo))
8  (ooo)
9  ((o)(o))
11 ((((o))))
13 ((o(o)))
16 (oooo)
17 (((oo)))
19 ((ooo))
23 (((o)(o)))
25 (((o))((o)))
27 ((o)(o)(o))
29 ((o((o))))
31 (((((o)))))

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A291442, A297571, A298478, A298534, A298536, A298537.

nn=500;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGweight[n_]:=If[n===1,1,1+Total[MGweight/@primeMS[n]]];
Select[Range[nn],SameQ@@MGweight/@primeMS[#]&]

A318690 Matula-Goebel numbers of powerful uniform rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 36, 49, 53, 59, 64, 67, 81, 83, 97, 100, 103, 121, 125, 127, 128, 131, 151, 196, 216, 225, 227, 241, 243, 256, 277, 289, 311, 331, 343, 361, 419, 431, 441, 484, 509, 512, 529, 541, 563, 625, 661, 691

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer n is a Matula-Goebel number of a powerful uniform rooted tree iff either n = 1 or n is a prime number whose prime index is a Matula-Goebel number of a powerful uniform rooted tree or n is a squarefree number taken to a power > 1 whose prime indices are all Matula-Goebel numbers of powerful uniform rooted trees.

			The sequence of all powerful uniform rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  32: (ooooo)
  36: (oo(o)(o))
  49: ((oo)(oo))

Gus Wiseman, The first 96 powerful uniform rooted trees.

Cf. A000081, A001694, A061775, A072774, A214577, A317705, A317707, A317710, A317717, A317719, A318611, A318612, A318689, A318692.

powunQ[n_]:=Or[n==1,If[PrimeQ[n],powunQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],Min@@FactorInteger[n][[All,2]]>1,And@@powunQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],powunQ]

A318692 Matula-Goebel numbers of series-reduced powerful uniform rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 512, 1024, 1444, 2048, 2401, 2744, 2809, 4096, 6859, 8192, 11236, 16384, 16807, 17161, 17689, 32768, 38416, 51529, 54872, 65536, 68644, 70756, 96721, 117649, 130321, 131072, 137641, 148877, 206116, 262144

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer n is a Matula-Goebel number of a series-reduced powerful uniform rooted tree iff either n = 1 or n is a squarefree number, whose prime indices are all Matula-Goebel numbers of series-reduced powerful uniform rooted trees, taken to a power > 1.

			The sequence of all series-reduced powerful uniform 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)
  196: (oo(oo)(oo))
  256: (oooooooo)
  343: ((oo)(oo)(oo))
  361: ((ooo)(ooo))
  512: (ooooooooo)

Cf. A000081, A001694, A061775, A072774, A214577, A291636, A317705, A317707, A317710, A318611, A318612, A318690, A318691.

srpowunQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],Min@@FactorInteger[n][[All,2]]>1,And@@srpowunQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100000],srpowunQ]

A322386 Numbers whose prime indices are not prime and already belong to the sequence.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 19, 28, 32, 38, 43, 49, 53, 56, 64, 76, 86, 98, 106, 107, 112, 128, 131, 133, 152, 163, 172, 196, 212, 214, 224, 227, 256, 262, 263, 266, 301, 304, 311, 326, 343, 344, 361, 371, 383, 392, 424, 428, 443, 448, 454, 512, 521, 524, 526, 532

Offset: 1

Gus Wiseman, Dec 05 2018

nonn

Union of A291636 (Matula-Goebel numbers of series-reduced rooted trees) and A322385.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiplicative semigroup: if x and y are in the sequence, then so is x*y. - Robert Israel, Dec 06 2018

			1 has no prime indices, so the definition is satisfied vacuously. - _Robert Israel_, Dec 07 2018
We have 301 = prime(4) * prime(14). Since 4 and 14 already belong to the sequence, so does 301.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000002, A001462, A007097, A079000, A079254, A214577, A276625, A291636, A304360, A320628, A322385.

Res:= 1: S:= {1}:
for n from 2 to 1000 do
  F:= map(numtheory:-pi, numtheory:-factorset(n));
  if F subset S then
    Res:= Res, n;
    if not isprime(n) then S:= S union {n} fi
fi
od:
Res; # Robert Israel, Dec 06 2018

tnpQ[n_]:=With[{m=PrimePi/@First/@If[n==1,{},FactorInteger[n]]},And[!MemberQ[m,_?PrimeQ],And@@tnpQ/@m]]
Select[Range[1000],tnpQ]

cp's OEIS Frontend

A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.

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A317719 Numbers that are not powerful tree numbers.

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A318993 Matula-Goebel number of the planted achiral tree determined by the n-th number whose consecutive prime indices are divisible.

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A358507 Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).

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A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

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A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.

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A298538 Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of nodes.

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A318690 Matula-Goebel numbers of powerful uniform rooted trees.

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A318692 Matula-Goebel numbers of series-reduced powerful uniform rooted trees.