A061775 -id:A061775 - cp's OEIS frontend

A299757 Weight of the strict integer partition with FDH number n.

0, 1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 5, 8, 6, 6, 9, 10, 7, 11, 7, 7, 8, 12, 6, 13, 9, 8, 8, 14, 7, 15, 10, 9, 11, 9, 9, 16, 12, 10, 8, 17, 8, 18, 10, 10, 13, 19, 11, 20, 14, 12, 11, 21, 9, 11, 9, 13, 15, 22, 9, 23, 16, 11, 12, 12, 10, 24, 13, 14, 10, 25, 10, 26, 17

Offset: 1

Gus Wiseman, Feb 18 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

In analogy with the Heinz number correspondence between integer partitions and positive integers (see A056239), FDH numbers give a correspondence between strict integer partitions and positive integers.

			Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9).

Cf. A004111, A050376, A056239, A061775, A064547, A106400, A213925, A215366, A246867, A279065, A279614, A299090, A299755, A299756, A299758, A299759.

FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[FDfactor[n]/.FDrules],{n,nn}]

A317713 Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.

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

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

			20 is the Matula-Goebel number of the tree (oo((o))), which has 4 distinct terminal subtrees: {(oo((o))), ((o)), (o), o}. So a(20) = 4.
See also illustrations in A061773.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Matula-Göbel numbers

One more than A324923.

Cf. A000081, A007097, A049076, A061773, A061775, A076146, A109082, A109129, A206491, A303431, A316476.

ids[n_]:=Union@@FixedPointList[Union@@(Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>PrimePi[p]]&/@#)&,{n}];
Table[Length[ids[n]],{n,100}]

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A324923(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #Set(lista); }; \\ Antti Karttunen, Oct 23 2023
A317713(n) = (1+A324923(n)); \\ Antti Karttunen, Oct 23 2023

a(n) = 1+A324923(n). - Antti Karttunen, Oct 23 2023

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

A061773 Triangle in which n-th row lists Matula-Goebel numbers for all rooted trees with n nodes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 15, 18, 20, 21, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 53, 59, 67, 25, 27, 30, 33, 35, 36, 39, 40, 42, 44, 46, 47, 48, 49, 51, 52, 56, 57, 58, 61, 62, 64, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 101, 106

Offset: 1

N. J. A. Sloane, Jun 22 2001

Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.
n-th row has A000081(n) terms.
First entry in row n is A005517(n).
Last entry in row n is A005518(n).
The Maple program yields row n after defining F = A005517(n) and L = A005518(n).

			The labels for the rooted trees with at most 4 nodes are as follows (x is the root):
                                         o
                                         |
               o         o        o   o  o
               |          \        \ /   |
     o  o   o  o  o o o    o   o    o    o
     |   \ /   |   \|/      \ /     |    |
  x  x    x    x    x        x      x    x
  1  2    4    3    8        6      7    5 (label)
Triangle begins:
1;
2;
3,4;
5,6,7,8;
9,10,11,12,13,14,16,17,19;
15,18,20,21,22,23,24,26,28,29,31,32,34,37,38,41,43,53,59,67;
25,27,30,33,35,36,39,40,42,44,46,47,48,49,51,52,56,57,58,61,62,64,68,\
71,73,74,76,79,82,83,86,89,101,106,107,109,118,127,131,134,139,157,163,\
179,191,241,277,331;
...
Triangle of rooted trees represented as finitary multisets begins:
(),
(()),
((())), (()()),
(((()))), (()(())), ((()())), (()()()),
((())(())), (()((()))), ((((())))), (()()(())), ((()(()))), (()(()())), (()()()()), (((()()))), ((()()())). - _Gus Wiseman_, Dec 21 2016

Alois P. Heinz, Rows n = 1..12, flattened
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Kevin Ryde, PARI/GP code calculating rows
Gus Wiseman, Matula-Goebel trees triangle illustration n=1..6
Index entries for sequences related to Matula-Goebel numbers
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A061775 (number of nodes), A000081 (row lengths), A005517 (row minimum), A005518 (row maximum), A214572 (row n=8).
Cf. A347620 (inverse permutation).
Cf. A276625, A279065, A279614.

n := 8: F := 45: L := 2221: with(numtheory): N := proc (m) local r, s: r := proc (m) options operator, arrow: op(1, factorset(m)) end proc: s := proc (m) options operator, arrow: m/r(m) end proc: if m = 1 then 1 elif bigomega(m) = 1 then 1+N(pi(m)) else N(r(m))+N(s(m))-1 end if end proc: A := {}: for k from F to L do if N(k) = n then A := `union`(A, {k}) else  end if end do: A;

F[n_] := F[n] = Which[n == 1, 1, n == 2, 2, Mod[n, 3] == 0, 3*5^(n/3-1), Mod[n, 3] == 1, 5^(n/3-1/3), True, 9*5^(n/3-5/3)]; L[n_] := L[n] = Switch[n, 1, 1, 2, 2, 3, 4, 4, 8, , Prime[L[n-1]]]; r[m] := FactorInteger[m][[1, 1]]; s[m_] := m/r[m]; NN[m_] := NN[m] = Which[m == 1, 1, PrimeOmega[m] == 1, 1+NN[PrimePi[m]], True, NN[r[m]]+NN[s[m]]-1]; row[n_] := Module[{A, k}, A = {}; For[k = F[n], k <= L[n], k++, If[NN[k] == n, A = Union[A, {k}]]]; A]; Table[row[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 06 2014, after Maple *)
nn=8;MGweight[n_]:=If[n===1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];
Take[GatherBy[Range[Switch[nn,1,1,2,2,3,4,,Nest[Prime,8,nn-4]]],MGweight],nn] (* _Gus Wiseman, Dec 21 2016 *)

PARI
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\\ See links.
```

More terms from Emeric Deutsch, May 01 2004

A324923 Number of distinct factors in the factorization of n into factors q(i) = prime(i)/i, i > 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 20 2019

nonn

Also the number of distinct proper terminal subtrees of the rooted tree with Matula-Goebel number n. See illustrations in A061773.

			The factorization 22 = q(1)^2 q(2) q(3) q(5) has four distinct factors, so a(22) = 4.

One less than A317713.
Cf. A000081, A003963, A061773, A061775, A109082, A109129, A120383, A196050.
Cf. A324850, A324922, A324924, A324925, A324931, A324934, A324935, A324936, A366385, A366386.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length[Union[difac[n]]],{n,100}]

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A324923(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #Set(lista); }; \\ Antti Karttunen, Oct 23 2023

a(n) = A317713(n) - 1.

a(n) = A196050(n) - A366386(n). - Antti Karttunen, Oct 23 2023

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

A302243 Total weight of the n-th twice-odd-factored multiset partition.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 03 2018

nonn

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. The n-th twice-odd-factored multiset partition is constructed by factoring 2n + 1 into prime numbers and then factoring each prime index into prime numbers and taking their prime indices.

			Sequence of multiset partitions begins: (), ((1)), ((2)), ((11)), ((1)(1)), ((3)), ((12)), ((1)(2)), ((4)), ((111)), ((1)(11)), ((22)), ((2)(2)), ((1)(1)(1)), ((13)), ((5)), ((1)(3)), ((2)(11)), ((112)), ((1)(12)), ((6)).

Cf. A003963, A007716, A034691, A048673, A056239, A061775, A063834 A064216, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302242.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Sum[PrimeOmega[k],{k,primeMS[2n-1]}],{n,100}]

a(n) = A302242(2n + 1).

A324924 Irregular triangle read by rows giving the factorization of n into factors q(i) = prime(i)/i, i > 0.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 1, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 4, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 4, 7, 1, 1, 1, 2, 2, 1, 1, 1, 8, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 2, 9

Offset: 1

Gus Wiseman, Mar 20 2019

Row n is the multiset of Matula-Goebel numbers of all proper terminal subtrees of the rooted tree with Matula-Goebel number n. For example, the rooted tree with Matula-Goebel number 1362 is (o(o)((oo)(oo))), with proper terminal subtrees {o,o,o,o,o,o,(o),(oo),(oo),((oo)(oo))}, which have Matula-Goebel numbers {1,1,1,1,1,1,2,4,4,49}, which is row 1362, as required.

			Triangle begins:
  {}
  1
  1  2
  1  1
  1  2  3
  1  1  2
  1  1  4
  1  1  1
  1  1  2  2
  1  1  2  3
  1  2  3  5
  1  1  1  2
  1  1  2  6
  1  1  1  4
  1  1  2  2  3
  1  1  1  1
  1  1  4  7
  1  1  1  2  2
  1  1  1  8
  1  1  1  2  3
  1  1  1  2  4
  1  1  2  3  5
  1  1  2  2  9
For example, row 65 is {1,1,1,2,2,3,6} because 65 = q(1)^3 q(2)^2 q(3) q(6).

Cf. A000081, A003963, A061775, A109082, A109129, A112798, A120383, A196050, A317713, A324850, A324922, A324923, A324925, A324931, A324934.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[difac[n],{n,30}]

A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

François Marques, Mar 14 2021

nonn

The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product_{T_i} prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration.)

			a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.

François Marques, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Matula-Goebel numbers

Other statistics are: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A358552 (node-height).
An ordered version is A358553.
Positions of first appearances are A358554.
A000081 counts rooted trees, ordered A000108.
A358575 counts rooted trees by nodes and internals.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A034781, A055277, A206487, A358576, A358578, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],[_],{0,Infinity}],{n,100}] (* Gus Wiseman, Nov 28 2022 *)

A342507(n) = if( n==1, 0, my(f=factor(n)); 1+sum(k=1,matsize(f)[1],A342507(primepi(f[k,1]))*f[k,2]));

a(1)=0 and a(n) = A061775(n) - A109129(n) for n > 1.

A206487 Number of ordered trees isomorphic (as rooted trees) to the rooted tree having Matula number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 4, 1, 3, 2, 6, 1, 1, 2, 2, 2, 6, 3, 2, 4, 4, 2, 6, 2, 3, 3, 2, 2, 5, 1, 3, 2, 6, 1, 4, 2, 4, 2, 4, 1, 12, 3, 2, 3, 1, 4, 6, 1, 3, 2, 6, 3, 10, 2, 6, 3, 3, 2, 12, 2, 5, 1, 4, 1, 12, 2, 4, 4, 4, 4, 12, 4, 3, 2, 4, 2, 6, 1, 3, 3, 6, 4, 6, 1, 8, 6, 2, 3, 10, 2, 6, 6, 5, 6, 6, 2, 6, 6, 2, 2, 20, 1, 6, 4, 3, 1, 12, 1, 1, 4, 12, 1, 12, 2, 2, 4, 4, 2, 6, 2, 12, 4, 6, 4, 15, 4, 4, 3, 9, 2, 12, 6, 4, 3, 6, 2, 24, 3, 4, 2, 6

Offset: 1

Emeric Deutsch, Apr 14 2012

nonn

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

a(n) = the number of times n occurs in A127301. - Antti Karttunen, Jan 03 2013

			a(4)=1 because the rooted tree with Matula number 4 is V and there is no other ordered tree isomorphic to V. a(6)=2 because the rooted tree corresponding to n = 6 is obtained by joining the trees A - B and C - D - E at their roots A and C. Interchanging their order, we obtain another ordered tree, isomorphic (as rooted tree) to the first one.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
P. Schultz, Enumeration of rooted trees with an application to group presentations, Discrete Math., 41, 1982, 199-214.
Index entries for sequences related to Matula-Goebel numbers

Cf. A127301.
Positions of 1's are 1 and A214577.
Positions of first appearances are A358507, unsorted A358508.
A000108 counts ordered rooted trees, unordered A000081.
A061775 and A196050 count nodes and edges in Matula-Goebel trees.
Cf. A001263, A003238, A014486, A358506.

with(numtheory): F := proc (n) options operator, arrow: factorset(n) end proc: PD := proc (n) local k, m, j: for k to nops(F(n)) do m[k] := 0: for j while is(n/F(n)[k]^j, integer) = true do m[k] := m[k]+1 end do end do: [seq([F(n)[q], m[q]], q = 1 .. nops(F(n)))] end proc: a := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n)) else mul(a(PD(n)[j][1])^PD(n)[j][2], j = 1 .. nops(F(n)))*factorial(add(PD(n)[k][2], k = 1 .. nops(F(n))))/mul(factorial(PD(n)[k][2]), k = 1 .. nops(F(n))) end if end proc: seq(a(n), n = 1 .. 160);

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}];
Table[treeperms[MGTree[n]],{n,100}] (* Gus Wiseman, Nov 21 2022 *)

a(1)=1; denoting by p(t) the t-th prime, if n = p(n_1)^{k_1}...p(n_r)^{k_r}, then a(n) = a(n_1)^{k_1}...a(n_r)^{k_r}*(k_1 + ... + k_r)!/[(k_1)!...(k_r)!] (see Theorem 1 in the Schultz reference, where the exponents k_j of N(n_j) have been inadvertently omitted).

A358371 Number of leaves in the n-th standard ordered rooted tree.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 13 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) = 3.

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

The triangle counting trees by this statistic is A001263, unordered A055277.
The version for unordered trees is A109129, nodes A061775, edges A196050.
The nodes are counted by A358372.
A000081 counts unordered rooted trees, ranked by A358378.
A000108 counts ordered rooted trees.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120
Cf. A004249, A005043, A063895, A187306, 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}]

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

			A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to Matula-Goebel numbers

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(|A106191(n)|) = A033844(n-1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
For standard instead of binary encoding we have A358506.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists binary encodings of ordered rooted trees.
Cf. A001263, A061775, A196050, A206487, A358505.

mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)

(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

A001222(a(n)) = A057515(n) for all n.

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