A317713 -id:A317713 - cp's OEIS frontend

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

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

A324936 Number of unlabeled rooted trees with n vertices whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 37, 83, 189, 436, 1014, 2373, 5578, 13156, 31104, 73665, 174665, 414427, 983606, 2334488

Offset: 1

Gus Wiseman, Mar 21 2019

The Matula-Goebel numbers of these trees are given by A324935.

			The a(1) = 1 through a(6) = 17 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o(o)))   (((ooo)))
                          (o((o)))   ((o)(oo))
                          ((((o))))  ((o(oo)))
                                     ((oo(o)))
                                     (o((oo)))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o(o))))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935.

durt[n_]:=Join@@Table[Select[Union[Sort/@Tuples[durt/@ptn]],UnsameQ@@Cases[#,{},{0,Infinity}]&],{ptn,IntegerPartitions[n-1]}];
Table[Length[durt[n]],{n,10}]

A317765 Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 3, 3, 4, 6, 4, 4, 5, 7, 2, 5, 5, 6, 8, 3, 6, 6, 7, 4, 9, 3, 4, 7, 7, 8, 4, 5, 10, 4, 3, 5, 8, 8, 9, 5, 6, 11, 5, 4, 6, 9, 9, 5, 10, 6, 7, 12, 6, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 7, 6, 8, 11, 11, 7, 12, 8, 9, 14, 4, 8, 7, 9, 12, 12, 3, 8

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.

nn=1000;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[Length[Union[Cases[exp[n],_,{0,Infinity},Heads->True]]],{n,100}]

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

A304486 Number of inequivalent leaf-colorings of the unlabeled rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 2, 2, 1, 4, 2, 4, 2, 5, 2, 4, 3, 4, 4, 2, 2, 7, 2, 5, 3, 9, 2, 5, 1, 7, 2, 4, 4, 9, 4, 7, 5, 7, 2, 11, 4, 4, 4, 4, 2, 12, 7, 4, 4, 11, 5, 7, 2, 16, 7, 5, 2, 11, 4, 2, 9, 11, 5, 5, 3, 9, 4, 11

Offset: 1

Gus Wiseman, Aug 17 2018

			Inequivalent representatives of the a(52) = 11 colorings of the tree (oo(o(o))) are the following.
  (11(1(1)))
  (11(1(2)))
  (11(2(1)))
  (11(2(2)))
  (11(2(3)))
  (12(1(1)))
  (12(1(2)))
  (12(1(3)))
  (12(3(1)))
  (12(3(3)))
  (12(3(4)))

Cf. A000081, A001678, A004111, A007097, A049076, A061773, A061775, A109082, A109129, A300626, A303024, A303431, A317713, A317994.

A325611 Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1.

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 7, 10, 10, 12, 12, 15, 12, 14, 16, 18, 14, 20, 16, 23, 20, 22, 22, 25, 25, 24, 23, 29, 26, 30, 27, 31, 33, 28, 32, 38, 36, 31, 36, 40, 37, 38, 33, 43, 44, 42, 39, 48, 39, 49, 45, 48, 43, 49, 49, 53, 47, 54, 47, 61

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)
Then a(n) is one plus the number of factors (counted with multiplicity) 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 12 nodes (o's plus brackets), so a(11) = 12.

Cf. A001222, A001221, A056239, A112798.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Mersenne numbers: A046051, A046800, A059305, A325610, A325612, A325625.

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

A325660 Number of ones in the q-signature of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 13 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)
Then a(n) is the number of factors of multiplicity one in the q-factorization of n.
Also the number of rooted trees appearing only once in the multiset of terminal subtrees of the rooted tree with Matula-Goebel number n.

Cf. A001694, A055231, A056169, A056239, A112798, A118914, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324968.
q-factorization: A324922, A324923, A324924, A325614, A325661.

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

A318049 Number of first/rest balanced rooted plane trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 3, 2, 6, 8, 11, 26, 28, 67, 96, 162, 316, 448, 922, 1435, 2572, 4660, 7563, 14397, 23896, 43337, 77097, 133071, 244787, 423093, 767732, 1367412, 2426612, 4408497, 7802348, 14152342, 25365035, 45602031, 82631362, 148246136, 269103870, 485379304

Offset: 1

Gus Wiseman, Aug 13 2018

nonn

A rooted plane tree is first/rest balanced if either (1) it is a single node, or (2a) the number of leaves in the first branch is equal to the number of branches minus one, and (2b) every branch is also first/rest balanced.

Also the number of composable free pure multifunctions (CPMs) with one atom and n positions. A CPM is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h and each of the g_i for i = 1, ..., k > 0 are CPMs, and the number of leaves in h is equal to k. The number of positions in a CPM is the number of brackets [...] plus the number of o's.

			The a(12) = 11 first/rest balanced rooted plane trees:
  (o(o(o((oo)oo))))
  (o(o((oo)(oo)o)))
  (o(o((oo)o(oo))))
  (o((oo)(o(oo))o))
  (o((oo)o(o(oo))))
  (o((oo)(oo)(oo)))
  ((oo)(o(o(oo)))o)
  ((oo)o(o(o(oo))))
  ((o(o(oo)))oooo)
  ((oo)(o(oo))(oo))
  ((oo)(oo)(o(oo)))
The a(12) = 11 composable free pure multifunctions:
  o[o[o[o[o][o,o]]]]
  o[o[o[o][o[o],o]]]
  o[o[o[o][o,o[o]]]]
  o[o[o][o[o[o]],o]]
  o[o[o][o,o[o[o]]]]
  o[o[o][o[o],o[o]]]
  o[o][o[o[o[o]]],o]
  o[o][o,o[o[o[o]]]]
  o[o][o[o[o]],o[o]]
  o[o][o[o],o[o[o]]]
  o[o[o[o]]][o,o,o,o]

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000081, A000108, A001003, A001006, A007853, A126120, A317713, A318046, A318048.

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

seq(n)={my(p=x*y+O(x^2)); for(n=1, n\2, p = x*y + x*sum(k=1, n, y^k * polcoef(p,k,y) * (O(x^(2*n-k+1)) + p)^k )); Vec(subst(p + O(x*x^n), y, 1)) } \\ Andrew Howroyd, Jan 22 2021

G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y + Sum_{k>=1} y^k * ([y^k] A(x,y)) * A(x,y)^k). - Andrew Howroyd, Jan 22 2021

Terms a(21) and beyond from Andrew Howroyd, Jan 22 2021

A324968 Matula-Goebel numbers of rooted identity trees whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 22, 26, 29, 31, 41, 58, 62, 79, 82, 101, 109, 127, 158, 179, 202, 218, 254, 271, 293, 358, 401, 421, 542, 547, 586, 599, 709, 802, 842, 929, 1063, 1094, 1198, 1231, 1361, 1418, 1609, 1741, 1858, 1913, 2126, 2411, 2462, 2722, 2749

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence ranks rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.

			The sequence of trees together with the Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    5: (((o)))
    6: (o(o))
   10: (o((o)))
   11: ((((o))))
   13: ((o(o)))
   22: (o(((o))))
   26: (o(o(o)))
   29: ((o((o))))
   31: (((((o)))))
   41: (((o(o))))
   58: (o(o((o))))
   62: (o((((o)))))
   79: ((o(((o)))))
   82: (o((o(o))))
  101: ((o(o(o))))
  109: (((o((o)))))
  127: ((((((o))))))

Gus Wiseman, The first 36 trees together with their Matula-Goebel numbers.

Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324969, A324970, A324978.

mgtree[n_Integer]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}],UnsameQ@@Cases[mgtree[#],{},{0,Infinity}]]&]

Intersection of A324935 and A276625.

A324970 Matula-Goebel numbers of rooted identity trees where not all terminal subtrees are different.

Original entry on oeis.org

15, 30, 33, 39, 47, 55, 65, 66, 78, 87, 93, 94, 110, 113, 123, 130, 137, 141, 143, 145, 155, 165, 167, 174, 186, 195, 205, 211, 226, 235, 237, 246, 257, 274, 282, 286, 290, 303, 310, 313, 317, 319, 327, 330, 334, 339, 341, 377, 381, 390, 395, 397, 403, 410

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.

			The sequence of trees together with the Matula-Goebel numbers begins:
   15: ((o)((o)))
   30: (o(o)((o)))
   33: ((o)(((o))))
   39: ((o)(o(o)))
   47: (((o)((o))))
   55: (((o))(((o))))
   65: (((o))(o(o)))
   66: (o(o)(((o))))
   78: (o(o)(o(o)))
   87: ((o)(o((o))))
   93: ((o)((((o)))))
   94: (o((o)((o))))
  110: (o((o))(((o))))
  113: ((o(o)((o))))
  123: ((o)((o(o))))
  130: (o((o))(o(o)))
  137: (((o)(((o)))))
  141: ((o)((o)((o))))
  143: ((((o)))(o(o)))
  145: (((o))(o((o))))
  155: (((o))((((o)))))
  165: ((o)((o))(((o))))
  167: (((o)(o(o))))
  174: (o(o)(o((o))))
  186: (o(o)((((o)))))
  195: ((o)((o))(o(o)))

Gus Wiseman, The first 36 trees together with their Matula-Goebel numbers.

Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324968, A324971, A324978.

mgtree[n_Integer]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}],!UnsameQ@@Cases[mgtree[#],{},{0,Infinity}]]&]

Complement of A324935 in A276625.

cp's OEIS Frontend

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

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A324936 Number of unlabeled rooted trees with n vertices whose non-leaf terminal subtrees are all different.

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A317765 Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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

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A304486 Number of inequivalent leaf-colorings of the unlabeled rooted tree with Matula-Goebel number n.

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A325611 Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1.

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A325660 Number of ones in the q-signature of n.

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A318049 Number of first/rest balanced rooted plane trees with n nodes.

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A324968 Matula-Goebel numbers of rooted identity trees whose non-leaf terminal subtrees are all different.