A324924 -id:A324924 - cp's OEIS frontend

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

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}]

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

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}]

A325614 Unsorted q-signature of n.

Original entry on oeis.org

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

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)
Row n lists the nonzero multiplicities in the q-factorization of n, in order of q-index. For example, row 11 is (1,1,1,1) and row 360 is (6,3,1).

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

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325615, A325660.

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

A324971 Number of rooted identity trees with n vertices whose non-leaf terminal subtrees are not all different.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 12, 31, 79, 192, 459, 1082, 2537, 5922, 13816, 32222, 75254, 176034, 412667, 969531, 2283278

Offset: 1

Gus Wiseman, Mar 21 2019

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

			The a(6) = 1 through a(8) = 12 trees:
  ((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)))))

The Matula-Goebel numbers of these trees are given by A324970.

Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935, A324936, A324979.

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

A325543 Width (number of leaves) of the rooted tree with Matula-Goebel number n!.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 7, 9, 12, 14, 16, 17, 20, 22, 25, 27, 31, 33, 36, 39, 42, 45, 47, 49, 53, 55, 58, 61, 65, 67, 70, 71, 76, 78, 81, 84, 88, 91, 95, 98, 102, 104, 108, 111, 114, 117, 120, 122, 127, 131, 134, 137, 141, 145, 149, 151, 156, 160, 163, 165, 169, 172

Offset: 0

Gus Wiseman, May 09 2019

nonn

Also the multiplicity of q(1) in the factorization of n! into factors q(i) = prime(i)/i. For example, the factorization of 7! is q(1)^9 * q(2)^3 * q(3) * q(4), so a(7) = 9.

			Matula-Goebel trees of the first 9 factorial numbers 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 leaves is the number of o's, which are (1, 1, 1, 2, 4, 5, 7, 9, 12, ...), 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, A325544.

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

For n > 1, a(n) = - 1 + Sum_{k = 1..n} A109129(k).

A325615 Sorted q-signature of n.

Original entry on oeis.org

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

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)
Row n is the multiset of nonzero multiplicities in the q-factorization of n. For example, row 11 is (1,1,1,1) and row 360 is (1,3,6).

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

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325614, A325660.

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

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

Original entry on oeis.org

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&]

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

cp's OEIS Frontend

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

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

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

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A325614 Unsorted q-signature of n.

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

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A324971 Number of rooted identity trees with n vertices whose non-leaf terminal subtrees are not all different.

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A325543 Width (number of leaves) of the rooted tree with Matula-Goebel number n!.

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A325615 Sorted q-signature of n.

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

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