A317710 -id:A317710 - cp's OEIS frontend

A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 8, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Sep 16 2018

nonn

			The a(144) = 17 factorizations:
  (144),
  (2*72), (3*48), (4*36), (6*24), (8*18), (9*16), (12*12),
  (2*3*24), (2*4*18), (2*6*12), (2*8*9), (3*4*12), (3*6*8),
  (2*2*6*6), (2*3*4*6), (3*3*4*4).

Cf. A001055, A045778, A047966, A052409, A072774, A296132, A303386, A317710, A317712, A317717, A317718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@Length/@Split[#]&]],{n,100}]

a(n) = Sum_{d|A052409(n)} A045778(n^(1/d)).

A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 16, 17, 19, 21, 31, 32, 53, 57, 59, 64, 67, 73, 85, 127, 128, 131, 133, 159, 241, 256, 269, 277, 311, 331, 335, 365, 367, 371, 393, 399, 439, 512, 649, 709, 719, 739, 751, 917, 933, 937, 1007, 1024, 1113, 1139, 1205, 1241, 1345, 1523

Offset: 1

Gus Wiseman, Jan 29 2019

nonn

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root. The only balanced rooted identity trees are rooted paths.

			The sequence of all unlabeled balanced rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  31: (((((o)))))
  32: (ooooo)
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  64: (oooooo)
  67: (((ooo)))
  73: (((o)(oo)))
  85: (((o))((oo)))

Cf. A000081, A004111, A007097, A048816, A184155, A276625, A306200, A306201, A306202, A316467, A317710.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n],1],And@@psidQ/@primeMS[n]];
mgtree[n_]:=If[n==1,{},mgtree/@primeMS[n]];
Select[Range[100],And[psidQ[#],SameQ@@Length/@Position[mgtree[#],{}]]&]

A317719 Numbers that are not powerful tree numbers.

Original entry on oeis.org

6, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 26, 28, 29, 30, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a powerful tree number iff either n = 1 or n is a prime number whose prime index is a powerful tree number, or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all powerful tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of numbers that are not powerful tree numbers together with their Matula-Goebel trees begins:
   6: (o(o))
  10: (o((o)))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Complement of A318612.
Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.
Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718.

powgoQ[n_]:=Or[n==1,If[PrimeQ[n],powgoQ[PrimePi[n]],And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!powgoQ[#]&]

A318691 Number of series-reduced powerful uniform rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 6, 1, 5, 4, 8, 1, 11, 1, 15, 6, 13, 1, 26, 3, 24, 9, 36, 1, 50, 1, 58, 14, 67, 7, 107, 1, 105, 25, 160, 1, 213, 1, 245, 45, 291, 1, 443, 5, 492, 68, 644, 1, 851, 15, 1019, 106, 1263, 1, 1785, 1, 1986, 189, 2592, 26, 3426, 1, 4071, 292

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A series-reduced powerful uniform rooted tree with n nodes is a powerful uniform multiset (all multiplicities are equal to the same number > 1) of series-reduced powerful uniform rooted trees with a total of n-1 nodes.

			The a(19) = 11 series-reduced powerful uniform rooted trees with 19 nodes:
  (((ooo)(ooo))((ooo)(ooo)))
  ((oo(oo)(oo))(oo(oo)(oo)))
  ((oo)(oo)(oo)(oo)(oo)(oo))
  ((oo)(oo)(ooooo)(ooooo))
  ((ooo)(ooo)(oooo)(oooo))
  (oo(oo)(oo)(oooo)(oooo))
  ((ooooo)(ooooo)(ooooo))
  (ooo(oooo)(oooo)(oooo))
  ((oooooooo)(oooooooo))
  (oo(ooooooo)(ooooooo))
  (oooooooooooooooooo)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000081, A001190, A001678, A001694, A003238, A072774, A317705, A317707, A317710, A318611, A318612, A318689, A318692.

rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],And[Min@@Length/@Split[#]>=2,SameQ@@Length/@Split[#]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=vector(n)); v[1]=1; for(n=1, n-1, my(u=WeighT(v[1..n])); v[n+1] = sumdiv(n,d,u[d]) - u[n]); v} \\ Andrew Howroyd, Dec 09 2020

a(p+1) = 1 for prime p. - Andrew Howroyd, Dec 09 2020

Terms a(51) and beyond from Andrew Howroyd, Dec 09 2020

A318689 Number of powerful uniform rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 12, 19, 23, 35, 36, 63, 64, 98, 112, 173, 174, 291, 292, 473, 509, 791, 792, 1345, 1356, 2158, 2257, 3634, 3635, 6053, 6054, 9807, 10091, 16173, 16216, 26783, 26784, 43076, 43880, 70631, 70632, 114975, 114976, 184665, 186996, 298644, 298645, 481978, 482011

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A powerful uniform rooted tree with n nodes is either a single powerful uniform branch with n-1 nodes, or a powerful uniform multiset (all multiplicities are equal to the same number > 1) of powerful uniform rooted trees with a total of n-1 nodes.

			The a(8) = 12 powerful uniform rooted trees:
  (((((((o)))))))
  ((((((oo))))))
  (((((o)(o)))))
  ((((o))((o))))
  (((((ooo)))))
  (((o)(o)(o)))
  ((((oooo))))
  (((oo)(oo)))
  ((oo(o)(o)))
  (((ooooo)))
  ((oooooo))
  (ooooooo)

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, All 35 powerful uniform rooted trees with 11 nodes.

Cf. A000081, A003238, A072774, A317705, A317707, A317710, A317717, A317718, A318611, A318612, A318690, A318691, A318692.

rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,And[Min@@Length/@Split[#]>=2,SameQ@@Length/@Split[#]]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,15}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=vector(n)); v[1]=1; for(n=1, n-1, my(u=WeighT(v[1..n])); v[n+1] = sumdiv(n,d,u[d]) - u[n] + v[n]); v} \\ Andrew Howroyd, Dec 09 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 09 2020

A318690 Matula-Goebel numbers of powerful uniform rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 36, 49, 53, 59, 64, 67, 81, 83, 97, 100, 103, 121, 125, 127, 128, 131, 151, 196, 216, 225, 227, 241, 243, 256, 277, 289, 311, 331, 343, 361, 419, 431, 441, 484, 509, 512, 529, 541, 563, 625, 661, 691

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer n is a Matula-Goebel number of a powerful uniform rooted tree iff either n = 1 or n is a prime number whose prime index is a Matula-Goebel number of a powerful uniform rooted tree or n is a squarefree number taken to a power > 1 whose prime indices are all Matula-Goebel numbers of powerful uniform rooted trees.

			The sequence of all powerful uniform rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  32: (ooooo)
  36: (oo(o)(o))
  49: ((oo)(oo))

Gus Wiseman, The first 96 powerful uniform rooted trees.

Cf. A000081, A001694, A061775, A072774, A214577, A317705, A317707, A317710, A317717, A317719, A318611, A318612, A318689, A318692.

powunQ[n_]:=Or[n==1,If[PrimeQ[n],powunQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],Min@@FactorInteger[n][[All,2]]>1,And@@powunQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],powunQ]

A318692 Matula-Goebel numbers of series-reduced powerful uniform rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 512, 1024, 1444, 2048, 2401, 2744, 2809, 4096, 6859, 8192, 11236, 16384, 16807, 17161, 17689, 32768, 38416, 51529, 54872, 65536, 68644, 70756, 96721, 117649, 130321, 131072, 137641, 148877, 206116, 262144

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer n is a Matula-Goebel number of a series-reduced powerful uniform rooted tree iff either n = 1 or n is a squarefree number, whose prime indices are all Matula-Goebel numbers of series-reduced powerful uniform rooted trees, taken to a power > 1.

			The sequence of all series-reduced powerful uniform rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   32: (ooooo)
   49: ((oo)(oo))
   64: (oooooo)
  128: (ooooooo)
  196: (oo(oo)(oo))
  256: (oooooooo)
  343: ((oo)(oo)(oo))
  361: ((ooo)(ooo))
  512: (ooooooooo)

Cf. A000081, A001694, A061775, A072774, A214577, A291636, A317705, A317707, A317710, A318611, A318612, A318690, A318691.

srpowunQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],Min@@FactorInteger[n][[All,2]]>1,And@@srpowunQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100000],srpowunQ]

A317720 Numbers that are not uniform relatively prime tree numbers.

Original entry on oeis.org

9, 12, 18, 20, 21, 23, 24, 25, 27, 28, 37, 39, 40, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 60, 61, 63, 65, 68, 69, 71, 72, 73, 74, 75, 76, 80, 81, 83, 84, 87, 88, 89, 90, 91, 92, 96, 97, 98, 99, 103, 104, 107, 108, 111, 112, 115, 116, 117, 120, 121, 122, 124

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform relatively prime tree number iff either n = 1 or n is a prime number whose prime index is a uniform relatively prime tree number, or n is a power of a squarefree number whose prime indices are relatively prime and are themselves uniform relatively prime tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of non-uniform tree numbers together with their Matula-Goebel trees begins:
   9: ((o)(o))
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  23: (((o)(o)))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  37: ((oo(o)))
  39: ((o)(o(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317590.

Cf. A317705, A317707, A317708, A317709, A317710, A317712, A317717, A317718.

rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[200],!rupQ[#]&]

cp's OEIS Frontend

A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

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A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.

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A317719 Numbers that are not powerful tree numbers.

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A318691 Number of series-reduced powerful uniform rooted trees with n nodes.

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A318689 Number of powerful uniform rooted trees with n nodes.

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Mathematica

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Extensions

A318690 Matula-Goebel numbers of powerful uniform rooted trees.

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Mathematica

A318692 Matula-Goebel numbers of series-reduced powerful uniform rooted trees.

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Mathematica

A317720 Numbers that are not uniform relatively prime tree numbers.

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