A109129 -id:A109129 - cp's OEIS frontend

A325609 Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i.

1, 2, 1, 4, 1, 5, 2, 1, 7, 3, 1, 9, 3, 1, 1, 12, 3, 1, 1, 14, 5, 1, 1, 16, 6, 2, 1, 17, 7, 3, 1, 1, 20, 8, 3, 1, 1, 22, 9, 3, 1, 1, 1, 25, 9, 3, 2, 1, 1, 27, 11, 4, 2, 1, 1, 31, 11, 4, 2, 1, 1, 33, 11, 4, 3, 1, 1, 1, 36, 13, 4, 3, 1, 1, 1, 39, 13, 4, 3, 1, 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 sequence of nonzero exponents in the q-factorization of n!.
Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n!.

			We have 10! = q(1)^16 q(2)^6 q(3)^2 q(4), so row n = 10 is (16,6,2,1).
Triangle begins:
  {}
   1
   2  1
   4  1
   5  2  1
   7  3  1
   9  3  1  1
  12  3  1  1
  14  5  1  1
  16  6  2  1
  17  7  3  1  1
  20  8  3  1  1
  22  9  3  1  1  1
  25  9  3  2  1  1
  27 11  4  2  1  1
  31 11  4  2  1  1
  33 11  4  3  1  1  1
  36 13  4  3  1  1  1
  39 13  4  3  1  1  1  1
  42 14  5  3  1  1  1  1

Row lengths are A000720.
Row sums are A325544(n) - 1.
Column k = 1 is A325543.
Cf. A056239, A067255, A112798, A118914, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
Factorial numbers: A000142, A011371, A022559, A071626, A115627, A325276.
q-factorization: A324922, A324923, A324924, A325614, 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,20}]

A325662 Matula-Goebel numbers of regular rooted stars.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 25, 27, 31, 32, 64, 81, 121, 125, 127, 128, 243, 256, 512, 625, 709, 729, 961, 1024, 1331, 2048, 2187, 3125, 4096, 5381, 6561, 8192, 14641, 15625, 16129, 16384, 19683, 29791, 32768, 52711, 59049, 65536, 78125, 131072, 161051

Offset: 1

Gus Wiseman, May 13 2019

nonn

Powers of members of A007097.
A regular rooted star is a rooted tree whose branches are all rooted paths of equal length.
The number of terms <= 10^k, k=0,1,2,...: 1, 7, 15, 26, 35, 46, 56, 67, 76, 87, 98, 109, 121, 131, 142, 154, 163, 175, 185, 198, 208, 220, 231, 241, 254, 265, 275, etc. - Robert G. Wilson v, May 13 2019

			The sequence of regular rooted stars together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    8: (ooo)
    9: ((o)(o))
   11: ((((o))))
   16: (oooo)
   25: (((o))((o)))
   27: ((o)(o)(o))
   31: (((((o)))))
   32: (ooooo)
   64: (oooooo)
   81: ((o)(o)(o)(o))
  121: ((((o)))(((o))))
  125: (((o))((o))((o)))
  127: ((((((o))))))
  128: (ooooooo)

Robert G. Wilson v, Table of n, a(n) for n = 1..275 (terms 1..48 from _Gus Wiseman_)

Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325663.

rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100],#==1||PrimePowerQ[#]&&rpQ[FactorInteger[#][[1,1]]]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[emax]; s = Union[s, s1], {k, 2, Length[ps]}]; s]; seq[10] (* Amiram Eldar, Jul 26 2024 *)

Sum_{n>=1} 1/a(n) = 1 + Product_{k>=1} 1/(A007097(k)-1) = 2.8928887669834086909... - Amiram Eldar, Jul 26 2024

A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 62, 64, 66, 72, 75, 80, 81, 88, 90, 93, 96, 99, 100, 108, 110, 120, 121, 124, 125, 127, 128, 132, 135, 144, 150, 155, 160, 162, 165, 176

Offset: 1

Gus Wiseman, May 13 2019

nonn

Products of members of A007097.

A rooted star is a rooted tree whose branches are all rooted paths.

			The sequence of rooted stars together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  15: ((o)((o)))
  16: (oooo)
  18: (o(o)(o))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))

Amiram Eldar, Table of n, a(n) for n = 1..10538 (terms up to A007097(12))

Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325662.

rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100],And@@rpQ/@First/@FactorInteger[#]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1, s2}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[0, emax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= pmax &]; s = Union[s, s2], {k, 2, Length[ps]}]; s]; seq[7] (* Amiram Eldar, Jul 26 2024 *)

Sum_{n>=1} 1/a(n) = Product_{k>=1} A007097(k)/(A007097(k)-1) = 4.30328607286382284593... . - Amiram Eldar, Jul 26 2024

A358728 Number of n-node rooted trees whose node-height is less than their number of leaves.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 10, 30, 76, 219, 582, 1662, 4614, 13080, 36903, 105098, 298689, 852734, 2434660, 6964349, 19931147, 57100177, 163647811, 469290004, 1346225668, 3863239150, 11089085961, 31838349956, 91430943515, 262615909503, 754439588007, 2167711283560

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Node-height is the number of nodes in the longest path from root to leaf.

			The a(1) = 0 through a(7) = 10 trees:
  .  .  .  (ooo)  (oooo)  (ooooo)   (oooooo)
                          ((oooo))  ((ooooo))
                          (o(ooo))  (o(oooo))
                          (oo(oo))  (oo(ooo))
                          (ooo(o))  (ooo(oo))
                                    (oooo(o))
                                    ((o)(ooo))
                                    ((oo)(oo))
                                    (o(o)(oo))
                                    (oo(o)(o))

Andrew Howroyd, Table of n, a(n) for n = 1..200

These trees are ranked by A358727.
For internals instead of node-height we have A358581, ordered A358585.
The case of equality is A358589 (square trees), ranked by A358577.
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.
Cf. A109082, A109129, A185650, A358552, A358582-A358586, A358587, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Depth[#]-1

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->sum(j=h+1, n-1, polcoef(p,j,y))), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

A358731 Matula-Goebel numbers of rooted trees whose number of nodes is one more than their node-height.

Original entry on oeis.org

4, 6, 7, 10, 13, 17, 22, 29, 41, 59, 62, 79, 109, 179, 254, 277, 293, 401, 599, 1063, 1418, 1609, 1787, 1913, 2749, 4397, 8527, 10762, 11827, 13613, 15299, 16519, 24859, 42043, 87803

Offset: 1

Gus Wiseman, Dec 01 2022

These are paths with a single extra leaf growing from them.
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.
Node-height is the number of nodes in the longest path from root to leaf.

			The terms together with their corresponding rooted trees begin:
    4: (oo)
    6: (o(o))
    7: ((oo))
   10: (o((o)))
   13: ((o(o)))
   17: (((oo)))
   22: (o(((o))))
   29: ((o((o))))
   41: (((o(o))))
   59: ((((oo))))
   62: (o((((o)))))
   79: ((o(((o)))))
  109: (((o((o)))))
  179: ((((o(o)))))
  254: (o(((((o))))))
  277: (((((oo)))))
  293: ((o((((o))))))
  401: (((o(((o))))))
  599: ((((o((o))))))

These trees are counted by A289207.
Positions of 1's 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. A206487, A209638, A316321, A358576, A358577, A358592, A358725.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Count[MGTree[#],_,{0,Infinity}]==Depth[MGTree[#]]&]

A196055 The terminal Wiener index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 6, 6, 4, 4, 4, 8, 8, 8, 5, 12, 8, 10, 12, 10, 10, 5, 10, 15, 6, 10, 12, 16, 10, 12, 5, 20, 6, 10, 12, 18, 15, 15, 12, 18, 10, 19, 16, 12, 14, 12, 12, 24, 20, 14, 12, 19, 20, 21, 7, 26, 18, 12, 10, 21, 18, 6, 22, 30, 14, 14, 15, 20, 14, 22, 18, 28, 19, 18, 16, 26, 14, 22, 12, 28, 24, 12, 12, 30, 14, 19, 14, 21, 24, 24

Offset: 1

Emeric Deutsch, Sep 29 2011

nonn

The terminal Wiener index of a connected graph is the sum of the distances between all pairs of nodes of degree 1.

The Matula-Goebel number of a rooted tree can be 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)=6 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2+2=6).

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, B. Furtula and M. Petrovic, Terminal Wiener index, J. Math. Chem., 46. 2009, 522-531.
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.
Index entries for sequences related to Matula-Goebel numbers

Cf. A109129, A196048, A196051, A348959.

with(numtheory): TW := proc (n) local r, s, LV, EPL, Tw: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LV := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then LV(pi(n)) else LV(r(n))+LV(s(n)) end if end proc: EPL := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then EPL(pi(n))+LV(pi(n)) else EPL(r(n))+EPL(s(n)) end if end proc: Tw := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then Tw(pi(n)) else Tw(r(n))+Tw(s(n))+EPL(r(n))*LV(s(n))+EPL(s(n))*LV(r(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = 1 then TW(pi(n))+LV(pi(n)) elif bigomega(n) = 1 then TW(pi(n))+EPL(n) else Tw(r(n))+Tw(s(n))+EPL(r(n))*LV(s(n))+EPL(s(n))*LV(r(n)) end if end proc; seq(TW(n), n = 1 .. 90);

Let LV(m) and EPL(m) denote the number of leaves and the external path length, respectively, of the rooted tree with Matula number m (see A109129 and A196048, where LV(m) and EPL(m) are obtained recursively). a(1)=0; if n=p(t) (=the t-th prime) and t is prime, then a(n) = a(t) + LV(t); if n=p(t) (=the t-th prime) and t is not prime, then a(n) = a(t) + LV(t) + EPL(t). Now assume that n is not prime; it can be written n=rs, where r is prime and s >= 2. If s is prime, then a(n) = a(r) - EPL(r) + a(s) - EPL(s) + EPL(r)*LV(s) + EPL(s)*LV(r); if s is not prime, then a(n) = a(r) - EPL(r) + a(s) + EPL(r)*LV(s) + EPL(s)*LV(r); the Maple program is based on this recursive formula.

If m > 2 then a(2^m) = m(m-1) because the rooted tree with Matula-Goebel number 2^m is a star with m edges and the vertices of each of the binomial(m,2) pairs of nodes of degree 1 are at distance 2.

A318048 Size of the span of the unlabeled rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 4, 4, 2, 6, 6, 5, 4, 6, 3, 9, 2, 6, 6, 4, 6, 6, 8, 10, 4, 12, 6, 10, 4, 9, 9, 6, 2, 12, 6, 9, 6, 6, 4, 9, 6, 9, 7, 6, 8, 15, 10, 15, 4, 5, 12, 9, 7, 4, 10, 16, 4, 7, 9, 8, 9, 10, 10, 11, 2, 13, 12, 6, 7, 14, 10, 9, 6, 10, 7, 21, 3, 12, 10, 12, 6

Offset: 1

Gus Wiseman, Aug 13 2018

nonn

The span of a tree is defined to be the set of possible terminal subtrees of initial subtrees, or, which is the same, the set of possible initial subtrees of terminal subtrees.

			42 is the Matula-Goebel number of (o(o)(oo)), which has span {o, (o), (oo), (ooo), (oo(oo)), (o(o)o), (o(o)(oo))}, so a(42) = 7.

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A007097, A007853, A049076, A061773, A061775, A109082, A109129, A206491, A317713, A318046.

ext[c_,{}]:=c;ext[c_,s:{}]:=Extract[c,s];rpp[c_,v_,{}]:=v;rpp[c_,v_,s:{}]:=ReplacePart[c,v,s];
RLO[ear_,rue:{}]:=Union@@(Function[x,rpp[ear,x,#2]]/@ReplaceList[ext[ear,#2],#1]&@@@Select[Tuples[{rue,Position[ear,_]}],MatchQ[ext[ear,#[[2]]],#[[1,1]]]&]);
RL[ear_,rue:{}]:=FixedPoint[Function[keeps,Union[keeps,Join@@(RLO[#,rue]&/@keeps)]],{ear}];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGTree[n_]:=If[n==1,{},MGTree/@primeMS[n]];
Table[Length[Union[Cases[RL[MGTree[n],{List[__List]:>List[]}],_List,{1,Infinity}]]],{n,100}]

A325608 Numbers whose factorization into factors prime(i)/i does not have weakly decreasing nonzero multiplicities.

Original entry on oeis.org

147, 245, 294, 357, 490, 511, 539, 588, 595, 637, 681, 714, 735, 845, 847, 853, 867, 903, 980, 1022, 1029, 1043, 1078, 1083, 1135, 1176, 1183, 1190, 1239, 1241, 1267, 1274, 1309, 1362, 1421, 1428, 1445, 1470, 1505, 1519, 1547, 1553, 1563, 1617, 1631, 1690

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, 147 = q(1)^5 q(2) q(4)^2 has multiplicities (5,1,2), which are not weakly decreasing, so 147 belongs to the sequence.

Cf. A001222, A056239, A112798, A118914.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
q-factorization: A324922, A324923, A324924, A324931, A325613, A325614, A325615, A325660, A325662.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[1000],!GreaterEqual@@Length/@Split[difac[#]]&]

A352288 Total cophenetic index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Kevin Ryde, Mar 16 2022

nonn

Mir, Rosselló, and Rotger, define the cophenetic value of a pair of childless vertices as the depth (distance down from the root) of their deepest common ancestor, and they then define the total cophenetic index of a tree as the sum of the cophenetic values over all such pairs.
a(n) = 0 iff n is in A325663, being rooted stars with any arm lengths, since the root (depth 0) is the common ancestor of every childless pair.
An identity relating the childless terminal Wiener index TW(n) = A348959(n) can be constructed by noting it measures distances from a pair of childless vertices to their common ancestor, and the cophenetic values measure from that ancestor up to the root.  So 2*a(n) + TW(n) is total depths Ext(n) = A196048(n) of the childless vertices, repeated by childless vertices C(n) = A109129(n) except itself, so that 2*a(n) + TW(n) = Ext(n)*(C(n) - 1)

			For n=111, the tree and its childless pairs and deepest common ancestors are
  root  R         pair  ancestor depth
       / \         G,D     A       1
      A   B        G,E     A       1
     /|\   \       D,E     A       1
    C D E   F     any,F    R       0
    |                             ---
    G                 total a(n) = 3

Arnau Mir, Francesc Rosselló, and Lucía Rotger, A New Balance Index for Phylogenetic Trees, arXiv:1202.1223 [q-bio.PE], 2012.
Kevin Ryde, PARI/GP Code
Index entries for sequences related to Matula-Goebel numbers

Cf. A348959 (terminal Wiener), A196048 (external length), A109129 (childless vertices).

Cf. A325663 (indices of 0's), A352289 (max by leaves).

PARI
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\\ See links.
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a(n) = Sum_{i=1..k} a(primepi(p[i])) + binomial(C(p[i]),2), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), and C(n) = A109129(n) is the number of childless vertices.

A358554 Least Matula-Goebel number of a rooted tree with n internal (non-leaf) nodes.

Original entry on oeis.org

1, 2, 3, 5, 11, 25, 55, 121, 275, 605, 1331, 3025, 6655, 14641, 33275, 73205

Offset: 1

Gus Wiseman, Nov 27 2022

Positions of first appearances in A342507.

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 rooted trees begin:
      1: o
      2: (o)
      3: ((o))
      5: (((o)))
     11: ((((o))))
     25: (((o))((o)))
     55: (((o))(((o))))
    121: ((((o)))(((o))))
    275: (((o))((o))(((o))))
    605: (((o))(((o)))(((o))))
   1331: ((((o)))(((o)))(((o))))
   3025: (((o))((o))(((o)))(((o))))
   6655: (((o))(((o)))(((o)))(((o))))
  14641: ((((o)))(((o)))(((o)))(((o))))
  33275: (((o))((o))(((o)))(((o)))(((o))))
  73205: (((o))(((o)))(((o)))(((o)))(((o))))

For height instead of internals we have A007097, firsts of A109082.
For leaves instead of internals we have A151821, firsts of A109129.
Positions of first appearances in A342507.
The ordered version gives firsts of A358553.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A001263, A206487, A358578, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[Count[MGTree[n],[_],{0,Infinity}],{n,1000}];
Table[Position[seq,n][[1,1]],{n,Union[seq]}]

cp's OEIS Frontend

A325609 Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i.

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A325662 Matula-Goebel numbers of regular rooted stars.

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A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

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A358728 Number of n-node rooted trees whose node-height is less than their number of leaves.

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A358731 Matula-Goebel numbers of rooted trees whose number of nodes is one more than their node-height.

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A196055 The terminal Wiener index of the rooted tree with Matula-Goebel number n.

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Maple

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A318048 Size of the span of the unlabeled rooted tree with Matula-Goebel number n.

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A325608 Numbers whose factorization into factors prime(i)/i does not have weakly decreasing nonzero multiplicities.

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