A324924 -id:A324924 - cp's OEIS frontend

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

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}]

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

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.

Original entry on oeis.org

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

A324979 Number of rooted trees with n vertices that are not identity trees but whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 29, 70, 168, 402, 959, 2284, 5434, 12923, 30727, 73055, 173678, 412830

Offset: 1

Gus Wiseman, Mar 21 2019

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.

			The a(3) = 1 through a(6) = 12 trees:
  (oo)  (ooo)   (oooo)    (ooooo)
        ((oo))  ((ooo))   ((oooo))
                (o(oo))   (o(ooo))
                (oo(o))   (oo(oo))
                (((oo)))  (ooo(o))
                          (((ooo)))
                          ((o)(oo))
                          ((o(oo)))
                          ((oo(o)))
                          (o((oo)))
                          (oo((o)))
                          ((((oo))))

The Matula-Goebel numbers of these trees are given by A324978.

Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935, A324936, A324970, A324971.

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

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[#]]&]

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

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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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A324979 Number of rooted trees with n vertices that are not identity trees but whose non-leaf terminal subtrees are all different.

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

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