A061775 -id:A061775 - cp's OEIS frontend

A298120 Matula-Goebel numbers of rooted trees in which all positive outdegrees are odd.

1, 2, 3, 5, 8, 11, 12, 18, 19, 20, 27, 30, 31, 32, 37, 44, 45, 48, 50, 61, 66, 67, 71, 72, 75, 76, 80, 99, 103, 108, 110, 113, 114, 120, 124, 125, 127, 128, 131, 148, 157, 162, 165, 171, 176, 180, 186, 190, 192, 193, 197, 200, 222, 223, 229, 242, 243, 244, 264

Offset: 1

Gus Wiseman, Jan 12 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
8  (ooo)
11 ((((o))))
12 (oo(o))
18 (o(o)(o))
19 ((ooo))
20 (oo((o)))
27 ((o)(o)(o))
30 (o(o)((o)))
31 (((((o)))))
32 (ooooo)
37 ((oo(o)))
44 (oo(((o))))
45 ((o)(o)((o)))
48 (oooo(o))
50 (o((o))((o)))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A007097, A026424, A061775, A214577, A276625, A277098, A290760, A291441, A291442, A291636, A297571, A298118.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otQ[n_]:=Or[n===1,With[{m=primeMS[n]},OddQ@Length@m&&And@@otQ/@m]];
Select[Range[1000],otQ]

A317711 Numbers that are not uniform tree numbers.

Original entry on oeis.org

12, 18, 20, 24, 28, 37, 40, 44, 45, 48, 50, 52, 54, 56, 60, 61, 63, 68, 71, 72, 74, 75, 76, 80, 84, 88, 89, 90, 92, 96, 98, 99, 104, 107, 108, 111, 112, 116, 117, 120, 122, 124, 126, 132, 135, 136, 140, 142, 144, 147, 148, 150, 152, 153, 156, 157, 160, 162

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform tree number iff either n = 1 or n is a power of a squarefree number whose prime indices are also uniform tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of non-uniform tree numbers together with their Matula-Goebel trees begins:
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  24: (ooo(o))
  28: (oo(oo))
  37: ((oo(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))
  48: (oooo(o))
  50: (o((o))((o)))
  52: (oo(o(o)))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  60: (oo(o)((o)))

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317590.

Cf. A317705, A317707, A317708, A317709, A317710 (complement), A317712, A317717, A317718.

rupQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100],!rupQ[#]&]

A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 10, 20, 36, 70, 134, 263, 513, 1022, 2030, 4076, 8203, 16614, 33738, 68833, 140796, 288989, 594621, 1226781, 2536532, 5256303, 10913196, 22700682, 47299699, 98714362, 206323140, 431847121, 905074333, 1899247187, 3990145833, 8392281473

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

Also the number of lone-child-avoiding rooted trees with n vertices and more than two branches.

			The a(4) = 1 through a(9) = 10 trees:
  (ooo)  (oooo)  (ooooo)   (oooooo)   (ooooooo)    (oooooooo)
                 (oo(oo))  (oo(ooo))  (oo(oooo))   (oo(ooooo))
                           (ooo(oo))  (ooo(ooo))   (ooo(oooo))
                                      (oooo(oo))   (oooo(ooo))
                                      (o(oo)(oo))  (ooooo(oo))
                                      (oo(o(oo)))  (o(oo)(ooo))
                                                   (oo(o(ooo)))
                                                   (oo(oo)(oo))
                                                   (oo(oo(oo)))
                                                   (ooo(o(oo)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The not necessarily lone-child-avoiding version is A331233.
The Matula-Goebel numbers of these trees are listed by A331490.
A000081 counts unlabeled rooted trees.
A001678 counts lone-child-avoiding rooted trees.
A001679 counts topologically series-reduced rooted trees.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A331489 lists Matula-Goebel numbers of series-reduced rooted trees.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]>2&&FreeQ[#,{_}]&]],{n,10}]

For n > 1, a(n) = A001679(n) - A001678(n).

a(37)-a(38) from Jinyuan Wang, Jun 26 2020

Terminology corrected (lone-child-avoiding, not series-reduced) by Gus Wiseman, May 10 2021

A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).
First differs from A331683 in having 49.
A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
Lone-child-avoiding means there are no unary branchings.
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 nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

			The sequence of all lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
The sequence of terms together with their prime indices begins:
     1: {}                  212: {1,1,16}
     4: {1,1}               214: {1,28}
     8: {1,1,1}             224: {1,1,1,1,1,4}
    14: {1,4}               256: {1,1,1,1,1,1,1,1}
    16: {1,1,1,1}           262: {1,32}
    28: {1,1,4}             304: {1,1,1,1,8}
    32: {1,1,1,1,1}         326: {1,38}
    38: {1,8}               343: {4,4,4}
    49: {4,4}               344: {1,1,1,14}
    56: {1,1,1,4}           361: {8,8}
    64: {1,1,1,1,1,1}       392: {1,1,1,4,4}
    76: {1,1,8}             424: {1,1,1,16}
    86: {1,14}              428: {1,1,28}
    98: {1,4,4}             448: {1,1,1,1,1,1,4}
   106: {1,16}              454: {1,49}
   112: {1,1,1,1,4}         512: {1,1,1,1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}     524: {1,1,32}
   152: {1,1,1,8}           526: {1,56}
   172: {1,1,14}            608: {1,1,1,1,1,8}
   196: {1,1,4,4}           622: {1,64}

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.

Not requiring local disjointness gives A291636.
Not requiring lone-child avoidance gives A316495.
A superset of A320269.
These trees are counted by A331680.
The semi-identity tree version is A331683.
The version containing 2 is A331873.
Cf. A001678, A007097, A050381, A061775, A196050, A302569, A302696, A316473, A316694, A316696, A316697, A331682, A331686, A331687, A331872, A331935.

msQ[n_]:=n==1||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000],msQ]

Intersection of A291636 and A316495.

A196047 Path length of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 5, 3, 6, 7, 10, 5, 8, 6, 9, 4, 9, 7, 7, 8, 8, 11, 11, 6, 12, 9, 9, 7, 12, 10, 15, 5, 13, 10, 11, 8, 10, 8, 11, 9, 13, 9, 11, 12, 12, 12, 15, 7, 10, 13, 12, 10, 9, 10, 16, 8, 10, 13, 14, 11, 13, 16, 11, 6, 14, 14, 12, 11, 14, 12, 14, 9, 14, 11, 15, 9, 15, 12, 17, 10, 12, 14, 17, 10, 15, 12, 15, 13, 12, 13, 13, 13, 18, 16, 13, 8, 19, 11, 16, 14

Offset: 1

Emeric Deutsch, Sep 27 2011

nonn

The path length of a rooted tree is defined as the sum of distances of all nodes to the root of the tree.

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(7) = 5 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (0+1+2+2 = 5).
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint 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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A049084, A020639.

Cf. A196048, A343006.

import Data.List (genericIndex)
a196047 n = genericIndex a196047_list (n - 1)
a196047_list = 0 : g 2 where
   g x = y : g (x + 1) where
     y = if t > 0 then a196047 t + a061775 t else a196047 r + a196047 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local r, s, N: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: N := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+N(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 100);

a[m_] := Module[{r, s, Nn},
   r[n_] := FactorInteger[n][[1, 1]];
   s[n_] := n/r[n];
   Nn[n_] := Which[n == 1, 1,
      PrimeOmega[n] == 1, 1+Nn[PrimePi[n]],
      True, Nn[r[n]]+Nn[s[n]]-1];
   Which[m == 1, 0,
   PrimeOmega[m] == 1, a[PrimePi[m]]+Nn[PrimePi[m]],
   True, a[r[m]]+a[s[m]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 03 2023, after Maple code *)

NPl(n) = { if(n==1, return([1,0]),
    my(f=factor(n)~, v=Mat(vector(#f,k,NPl(primepi(f[1,k]))~))  );
    return( [ 1+sum(k=1,#f,v[1,k]*f[2,k]) , sum(k=1,#f,(v[1,k]+v[2,k])*f[2,k]) ] ) )
  };
A196047(n) = NPl(n)[2]; \\ François Marques, Apr 02 2021

a(1)=0; if n=prime(t) then a(n)=a(t)+N(t), where N(t) is the number of nodes of the rooted tree with Matula number t; if n=r*s (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.

a(n) = A196048(n) + A343006(n). - François Marques, Apr 02 2021

A279614 a(1)=1, a(d(x_1)..d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 5, 5, 6, 7, 6, 6, 7, 7, 6, 7, 6, 8, 8, 8, 8, 7, 7, 7, 7, 7, 9, 8, 8, 8, 7, 9, 8, 10, 8, 7, 9, 8, 9, 8, 9, 7, 10, 9, 8, 9, 8, 9, 8, 9, 9, 9, 8, 11, 10, 10, 9, 9, 10, 8, 9, 10, 9, 10, 10, 8, 10, 9, 11, 8, 9, 8, 8, 9, 11, 12, 9, 8, 10, 10, 9

Offset: 1

Gus Wiseman, Dec 15 2016

nonn

A Fermi-Dirac prime (A050376) is a positive integer of the form p^(2^k) where p is prime and k>=0.

In analogy with the Matula-Goebel correspondence between rooted trees and positive integers (see A061775), the iterated normalized Fermi-Dirac representation gives a correspondence between rooted identity trees and positive integers. Then a(n) is the number of nodes in the rooted identity tree corresponding to n.

			Sequence of rooted identity trees represented as finitary sets begins:
{}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}{{}}}, {{{{{{}}}}}},
{{}{{{}}}}, {{{}{{}}}}, {{}{{{{}}}}}, {{{{{{{}}}}}}}, {{{}}{{{}}}},
{{{}{{{}}}}}, {{}{{{{{}}}}}}, {{{}}{{{{}}}}}, {{{{}{{}}}}},
{{{}{{{{}}}}}}, {{}{{}{{}}}}, {{{{{{{{}}}}}}}}, {{{{}}}{{{{}}}}},
{{{}}{{{{{}}}}}}, {{}{{{{{{}}}}}}}, {{{{}}{{{}}}}}, {{}{{}}{{{}}}}.

OEIS Wiki, "Fermi-Dirac representation" of n

Cf. A004111, A050376, A061773, A061775, A084400, A276625, A279065.

nn=200;
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)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];
FDweight[n_?(#<=nn&)]:=If[n===1,1,1+Total[FDweight[Position[FDprimeList,#][[1,1]]]&/@FDfactor[n]]];
Array[FDweight,nn]

Number of appearances of n is |a^{-1}(n)| = A004111(n).

A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 32, 36, 38, 42, 48, 54, 56, 64, 72, 76, 78, 84, 96, 98, 106, 108, 112, 114, 126, 128, 144, 152, 156, 162, 168, 192, 196, 212, 216, 222, 224, 228, 234, 252, 256, 262, 266, 288, 294, 304, 312, 318, 324, 336, 342, 366, 378

Offset: 1

Gus Wiseman, Aug 20 2018

nonn

A number x is totally transitive if (1) whenever prime(y) divides x it follows that y is totally transitive and (2) if prime(y) divides x and prime(z) divides y then prime(z) also divides x.

			The sequence of all totally transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))
  98: (o(oo)(oo))

Cf. A000081, A001678, A004111, A007097, A061775, A276625, A279861, A290689, A290760, A290822, A291636, A318185, A318187.

subprimes[n_]:=If[n==1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
trmgQ[n_]:=Or[n==1,And[Divisible[n,Times@@subprimes[n]],And@@Cases[FactorInteger[n],{p_,_}:>trmgQ[PrimePi[p]]]]];
Select[Range[100],trmgQ]

A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 4, 5, 3, 6, 7, 4, 5, 7, 6, 7, 11, 7, 7, 9, 10, 7, 13, 7, 11, 9, 11, 11, 13, 11, 12, 15, 16, 10, 19, 19, 15, 18, 16, 16, 18, 10, 18, 18, 17, 15, 21, 15, 18, 24, 23, 19, 23, 25, 25, 18, 26, 25, 28, 21, 21, 25, 23, 21, 29, 28, 31, 21, 24, 23

Offset: 1

Gus Wiseman, May 12 2019

nonn

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
For n > 1, a(n) is the multiplicity of q(1) = 2 in the q-factorization of 2^n - 1.

			The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 6 leaves (o's), so a(11) = 6.

Keith Briggs, Matula numbers and rooted trees.

Cf. A001222, A001221, A056239, A112798.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325625.

mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];
Table[mglv[2^n-1],{n,30}]

More terms from Jinyuan Wang, Feb 25 2025

A331490 Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

Original entry on oeis.org

8, 16, 28, 32, 56, 64, 76, 98, 112, 128, 152, 172, 196, 212, 224, 256, 266, 304, 343, 344, 392, 424, 428, 448, 512, 524, 532, 602, 608, 652, 686, 688, 722, 742, 784, 848, 856, 896, 908, 931, 1024, 1048, 1052, 1064, 1204, 1216, 1244, 1304, 1372, 1376, 1444

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

We say that a rooted tree is (topologically) series-reduced if no vertex has degree 2.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.

			The sequence of all series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:
    8: (ooo)
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   98: (o(oo)(oo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
  212: (oo(oooo))
  224: (ooooo(oo))
  256: (oooooooo)
  266: (o(oo)(ooo))
  304: (oooo(ooo))
  343: ((oo)(oo)(oo))
  344: (ooo(o(oo)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

These trees are counted by A331488.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[1000],PrimeOmega[#]>2&&srQ[#]&]

A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

Original entry on oeis.org

1, 4, 8, 16, 18, 25, 32, 36, 50, 57, 64, 72, 100, 114, 121, 128, 137, 144, 200, 228, 242, 249, 256, 258, 274, 281, 288, 385, 393, 400, 456, 484, 498, 505, 512, 516, 548, 562, 569, 576, 770, 786, 793, 800, 897, 905, 912, 968, 996, 1010, 1017, 1024, 1032, 1096

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 initial terms and their corresponding trees:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   18: ((oo)o)
   25: (o(oo))
   32: (ooooo)
   36: ((oo)oo)
   50: (o(oo)o)
   57: (oo(oo))
   64: (oooooo)
   72: ((oo)ooo)
  100: (o(oo)oo)
  114: (oo(oo)o)
  121: (ooo(oo))
  128: (ooooooo)
  137: ((oo)(oo))
  144: ((oo)oooo)
  200: (o(oo)ooo)

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

These trees are counted by A005043.
The series-reduced case appears to be counted by A284778.
The unordered version is A291636, counted by A001678.
A000081 counts unlabeled rooted trees, ranked by A358378.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120.
Cf. A000014, A001263, A001679, A004249, A061775, A063895, A187306, A331489, A331490, A331934, A358373, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],FreeQ[srt[#],[_]?(Length[#]==1&)]&]

cp's OEIS Frontend

A298120 Matula-Goebel numbers of rooted trees in which all positive outdegrees are odd.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A317711 Numbers that are not uniform tree numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A196047 Path length of the rooted tree with Matula-Goebel number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A279614 a(1)=1, a(d(x_1)*..*d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

Views

Author

Keywords

A279614 a(1)=1, a(d(x_1)..d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.