A109129 -id:A109129 - cp's OEIS frontend

A325661 q-powerful numbers. Numbers whose factorization into factors prime(i)/i has no factor of multiplicity 1.

1, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 75, 81, 98, 100, 108, 121, 125, 128, 144, 150, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 300, 324, 338, 343, 361, 363, 375, 392, 400, 432, 441, 450, 484, 486, 500, 507, 512, 529, 576

Offset: 1

Gus Wiseman, May 13 2019

nonn

First differs from A070003 in having 1 and lacking 147.
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)
Also Matula-Goebel numbers of rooted trees with no terminal subtree appearing at only one place in the tree.

			The sequence of terms together with their q-signatures begins:
    1: {}
    4: {2}
    8: {3}
    9: {2,2}
   16: {4}
   18: {3,2}
   25: {2,2,2}
   27: {3,3}
   32: {5}
   36: {4,2}
   49: {4,2}
   50: {3,2,2}
   54: {4,3}
   64: {6}
   72: {5,2}
   75: {3,3,2}
   81: {4,4}
   98: {5,2}
  100: {4,2,2}

Charlie Neder, Table of n, a(n) for n = 1..1071 (Terms <= 100000)

Cf. A001222, A001221, A001694, A056239, A112798, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713.
q-factorization: A324922, A324923, A324924, A325615, A325660.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[100],Count[Length/@Split[difac[#]],1]==0&]

A358725 Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.

Original entry on oeis.org

9, 15, 18, 21, 23, 25, 27, 30, 33, 35, 36, 39, 42, 45, 46, 47, 49, 50, 51, 54, 55, 57, 60, 61, 63, 65, 66, 69, 70, 72, 73, 75, 77, 78, 81, 83, 84, 85, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 108, 110, 111, 113, 114, 115, 117, 119, 120, 121

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges 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.

			The terms together with their corresponding trees begin:
   9: ((o)(o))
  15: ((o)((o)))
  18: (o(o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))
  33: ((o)(((o))))
  35: (((o))(oo))
  36: (oo(o)(o))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  45: ((o)(o)((o)))
  46: (o((o)(o)))
  47: (((o)((o))))
  49: ((oo)(oo))
  50: (o((o))((o)))

Gus Wiseman, The first 64 ordered trees with a greater number of internal vertices than edge-height.

Complement of A209638 (the case of equality).
These trees are counted by A316321.
Positions of positive terms in A358724.
The case of equality for node-height is A358576.
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.
Differences: A358580, A358724, A358726, A358729.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A185650, A206487, A358577, A358578, A358581-A358586, A358587, A358592, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],[_],{0,Infinity}]>Depth[MGTree[#]]-2&]

A342507(a(n)) > A109082(a(n)).

A358730 Positions of first appearances in A358729 (number of nodes minus node-height).

Original entry on oeis.org

1, 4, 8, 16, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049

Offset: 1

Gus Wiseman, Dec 01 2022

First differs from A334198 in having 13122 instead of 12005.
Node-height is the number of nodes in the longest path from root to leaf.
After initial terms, this appears to become A038754.

			The terms together with their corresponding rooted trees begin:
      1: o
      4: (oo)
      8: (ooo)
     16: (oooo)
     27: ((o)(o)(o))
     54: (o(o)(o)(o))
     81: ((o)(o)(o)(o))
    162: (o(o)(o)(o)(o))
    243: ((o)(o)(o)(o)(o))
    486: (o(o)(o)(o)(o)(o))
    729: ((o)(o)(o)(o)(o)(o))

Positions of first appearances in A358729.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves.
MG differences: A358580, A358724, A358726, A358729.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A080936, A206487, A209638, A316321, A358576, A358577, A358592, A358725, A358731.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
rd=Table[Count[MGTree[n],_,{0,Infinity}]-(Depth[MGTree[n]]-1),{n,10000}];
Table[Position[rd,k][[1,1]],{k,Union[rd]}]

A366385 Divide n by its largest prime factor, then multiply with the index of that same prime; a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 4, 6, 6, 5, 8, 6, 8, 9, 8, 7, 12, 8, 12, 12, 10, 9, 16, 15, 12, 18, 16, 10, 18, 11, 16, 15, 14, 20, 24, 12, 16, 18, 24, 13, 24, 14, 20, 27, 18, 15, 32, 28, 30, 21, 24, 16, 36, 25, 32, 24, 20, 17, 36, 18, 22, 36, 32, 30, 30, 19, 28, 27, 40, 20, 48, 21, 24, 45, 32, 35, 36, 22, 48, 54, 26, 23, 48, 35

Offset: 1

Antti Karttunen, Oct 23 2023

nonn

Cf. A000720, A006530, A052126, A061395.
Cf. A196050 (number of iterations needed to reach 1), A366388 (number of iterations to reach the nearest power of 2), A109129 (exponent of the nearest power of 2 reached).
Cf. also A366387, A324923.

Array[PrimePi[#2]*#1/#2 & @@ {#, FactorInteger[#][[-1, 1]]} &, 85] (* Michael De Vlieger, Oct 23 2023 *)

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A366385(n) = { my(gpf=A006530(n)); primepi(gpf)*(n/gpf); };

a(n) = A052126(n)*A061395(n) = (n/A006530(n)) * A000720(A006530(n)).

A318046 a(n) is the number of initial subtrees (subtrees emanating from the root) of the unlabeled rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 2, 5, 4, 5, 3, 4, 3, 7, 2, 4, 5, 3, 4, 5, 5, 6, 3, 10, 4, 9, 3, 5, 7, 6, 2, 9, 4, 7, 5, 4, 3, 7, 4, 5, 5, 4, 5, 13, 6, 8, 3, 5, 10, 7, 4, 3, 9, 13, 3, 5, 5, 5, 7, 6, 6, 9, 2, 10, 9, 4, 4, 11, 7, 5, 5, 6, 4, 19, 3, 9, 7, 6, 4, 17, 5, 7, 5

Offset: 1

Gus Wiseman, Aug 13 2018

nonn

We require that an initial subtree contain either all or none of the branchings under any given node.

			70 is the Matula-Goebel number of the tree (o((o))(oo)), which has 7 distinct initial subtrees: {o, (ooo), (oo(oo)), (o(o)o), (o(o)(oo)), (o((o))o), (o((o))(oo))}. So a(70) = 7.

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A007097, A007853, A049076, A061773, A061775, A076146, A109082, A109129, A206491, A303431, A316476, A317713.

si[n_]:=If[n==1,1,1+Product[si[PrimePi[b[[1]]]]^b[[2]],{b,FactorInteger[n]}]];
Array[si,100]

a(1) = 1 and if n > 1 has prime factorization n = prime(x_1)^y_1 * ... * prime(x_k)^y_k then a(n) = 1 + a(x_1)^y_1 * ... * a(x_k)^y_k.

A324933 Denominator in the division of n by the product of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 21 2019

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.

			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, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

The numerators are A324932.

Cf. A003963, A109129, A120383, A196050, A317713, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

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

A325544 Number of nodes in the rooted tree with Matula-Goebel number n!.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 30, 34, 38, 42, 47, 51, 55, 60, 64, 69, 74, 79, 84, 89, 95, 100, 106, 111, 116, 122, 127, 132, 138, 143, 149, 155, 160, 165, 171, 177, 182, 188, 193, 199, 206, 212, 218, 224, 230, 237, 243, 249, 254, 261, 268, 274, 280

Offset: 0

Gus Wiseman, May 09 2019

nonn

Also one plus the number of factors in the factorization of n! into factors q(i) = prime(i)/i. For example, the q-factorization of 7! is 7! = q(1)^9 * q(2)^3 * q(3) * q(4), with 14 = a(7) - 1 factors.

			Matula-Goebel trees of the first 9 factorial number are:
  0!: o
  1!: o
  2!: (o)
  3!: (o(o))
  4!: (ooo(o))
  5!: (ooo(o)((o)))
  6!: (oooo(o)(o)((o)))
  7!: (oooo(o)(o)((o))(oo))
  8!: (ooooooo(o)(o)((o))(oo))
The number of nodes is the number of o's plus the number of brackets, giving {1,1,2,4,6,9,12,15,18}, as required.

Cf. A000081, A001222, A056239, A324922, A324923, A324924.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Factorial numbers: A000142, A011371, A022559, A071626, A076934, A115627, A325272, A325273, A325276, A325508, A325543.

mgwt[n_]:=If[n==1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>mgwt[PrimePi[p]]*k]]];
Table[mgwt[n!],{n,0,100}]

For n > 1, a(n) = 1 - n + Sum_{k = 1..n} A061775(k).

A325613 Full q-signature of n. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the q-factorization of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 12 2019

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)
Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n.

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  2 1
  2 0 0 1
  3
  2 2
  2 1 1
  1 1 1 0 1
  3 1
  2 1 0 0 0 1
  3 0 0 1
  2 2 1
  4
  2 0 0 1 0 0 1
  3 2
  3 0 0 0 0 0 0 1
  3 1 1

Row lengths are A061395.
Row sums are A196050.
Row-maxima are A109129.
The number whose full prime signature is the n-th row is A324922(n).
Cf. A067255.
Matula-Goebel numbers: A007097, A061775, A109082, A317713.
q-factorization: A324923, A324924, A325613, A325614, A325615, A325660.

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

A348959 Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 2, 0, 3, 2, 6, 4, 4, 0, 8, 3, 8, 5, 12, 2, 10, 6, 10, 10, 5, 4, 15, 6, 10, 12, 16, 4, 12, 0, 20, 6, 10, 12, 18, 8, 15, 12, 18, 3, 19, 8, 12, 14, 12, 5, 24, 20, 14, 12, 19, 12, 21, 7, 26, 18, 12, 2, 21, 10, 6, 22, 30, 14, 14, 6, 20, 14, 22, 10, 28, 10

Offset: 1

Kevin Ryde, Nov 05 2021

nonn

This is a variation on the terminal Wiener index defined by Gutman, Furtula, and Petrović. Here terminal vertices are taken as the childless vertices, so a(n) is the sum of the path lengths between pairs of childless vertices.

This sequence differs from the free tree form A196055 when n is prime, since n prime means the root is degree 1 so is a terminal vertex for A196055 but not here.

Kevin Ryde, Table of n, a(n) for n = 1..10000
F. Goebel, On a 1-1-Correspondence between Rooted Trees and Natural Numbers, Journal of Combinatorial Theory, series B, volume 29, 1980, pages 141-143.
Ivan Gutman, Boris Furtula and Miroslav Petrović, Terminal Wiener Index, Journal of Mathematical Chemistry, volume 46, 2009, pages 522-531.
D. W. Matula, A Natural Rooted Tree Enumeration By Prime Factorization, SIAM Review, volume 10, number 2, April 1968, page 273 (also at JSTOR).
Kevin Ryde, PARI/GP Code and Notes
Index entries for sequences related to Matula-Goebel numbers

Cf. A196055 (free tree), A196048 (external path length), A109129 (childless vertices), A288469 (unplant).

Cf. A027746 (prime factorization).

PARI
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a(n) = Sum_{j=1..k} (a(primepi(p[j])) + E(p[j])*(C(n)-C(p[j]))), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), E(n) = A196048(n) is external path length, and C(n) = A109129(n) is number of childless vertices.
a(n) = A196055(n) - (A196048(n) if n prime).
a(n) = A196055(A288469(n)).

A324932 Numerator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 13, 27, 7, 29, 5, 31, 32, 33, 34, 35, 9, 37, 19, 13, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 26, 53, 27, 11, 14, 57, 29, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 23

Offset: 1

Gus Wiseman, Mar 21 2019

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.

			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, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

The denominators are A324933.

Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

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

cp's OEIS Frontend

A325661 q-powerful numbers. Numbers whose factorization into factors prime(i)/i has no factor of multiplicity 1.

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A358725 Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.

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A358730 Positions of first appearances in A358729 (number of nodes minus node-height).

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A366385 Divide n by its largest prime factor, then multiply with the index of that same prime; a(1) = 0 by convention.

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A318046 a(n) is the number of initial subtrees (subtrees emanating from the root) of the unlabeled rooted tree with Matula-Goebel number n.

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A324933 Denominator in the division of n by the product of prime indices of n.

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A325544 Number of nodes in the rooted tree with Matula-Goebel number n!.

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A325613 Full q-signature of n. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the q-factorization of n.

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