A306021 -id:A306021 - cp's OEIS frontend

A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 49, 53, 64, 81, 113, 128, 131, 151, 161, 165, 169, 225, 243, 256, 311, 343, 361, 512, 719, 729, 1024, 1291, 1321, 1619, 1937, 1957, 2021, 2048, 2093, 2117, 2187, 2197, 2257, 2401, 2805, 2809, 3375, 3671, 4096, 6561

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform, regular, and spans an initial interval of positive integers, so its MM-number 15463 belongs to the sequence.

			The sequence of all uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
    1: {}
    2: {{}}
    3: {{1}}
    4: {{},{}}
    7: {{1,1}}
    8: {{},{},{}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   16: {{},{},{},{}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   32: {{},{},{},{},{}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   64: {{},{},{},{},{},{}}
   81: {{1},{1},{1},{1}}
  113: {{1,2,3}}
  128: {{},{},{},{},{},{},{}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  165: {{1},{2},{3}}
  169: {{1,2},{1,2}}
  225: {{1},{1},{2},{2}}
  243: {{1},{1},{1},{1},{1}}
  256: {{},{},{},{},{},{},{},{}}
  311: {{1,1,1,1,1,1}}
  343: {{1,1},{1,1},{1,1}}
  361: {{1,1,1},{1,1,1}}
  512: {{},{},{},{},{},{},{},{},{}}
  719: {{1,1,1,1,1,1,1}}
  729: {{1},{1},{1},{1},{1},{1}}

Cf. A005176, A007016, A112798, A302242, A306021, A319056, A319189, A320324, A321698, A321717, A322554, A322703, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 7, 13, 15, 19, 53, 113, 131, 151, 161, 165, 311, 719, 1291, 1321, 1619, 1937, 1957, 2021, 2093, 2117, 2257, 2805, 3671, 6997, 8161, 10627, 13969, 13987, 14023, 15617, 17719, 17863, 20443, 22207, 22339, 38873, 79349, 84017, 86955, 180503, 202133

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, regular if all vertices appear the same number of times, and strict if there are no repeated parts. For example, {{1,1},{2,3},{2,3}} is uniform and regular but not strict, so its MM-number 15463 does not belong to the sequence. Note that the parts of parts such as {1,1} do not have to be distinct, only the multiset of parts.

			The sequence of all strict uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
      1: {}
      2: {{}}
      3: {{1}}
      7: {{1,1}}
     13: {{1,2}}
     15: {{1},{2}}
     19: {{1,1,1}}
     53: {{1,1,1,1}}
    113: {{1,2,3}}
    131: {{1,1,1,1,1}}
    151: {{1,1,2,2}}
    161: {{1,1},{2,2}}
    165: {{1},{2},{3}}
    311: {{1,1,1,1,1,1}}
    719: {{1,1,1,1,1,1,1}}
   1291: {{1,2,3,4}}
   1321: {{1,1,1,2,2,2}}
   1619: {{1,1,1,1,1,1,1,1}}
   1937: {{1,2},{3,4}}
   1957: {{1,1,1},{2,2,2}}
   2021: {{1,4},{2,3}}
   2093: {{1,1},{1,2},{2,2}}
   2117: {{1,3},{2,4}}
   2257: {{1,1,2},{1,2,2}}
   2805: {{1},{2},{3},{4}}
   3671: {{1,1,1,1,1,1,1,1,1}}
   6997: {{1,1,2,2,3,3}}
   8161: {{1,1,1,1,1,1,1,1,1,1}}
  10627: {{1,1,1,1,2,2,2,2}}
  13969: {{1,2,2},{1,3,3}}
  13987: {{1,1,3},{2,2,3}}
  14023: {{1,1,2},{2,3,3}}
  15617: {{1,1},{2,2},{3,3}}
  17719: {{1,2},{1,3},{2,3}}
  17863: {{1,1,1,1,1,1,1,1,1,1,1}}

Cf. A005117, A007016, A112798, A302242, A306017, A306021, A319056, A319189, A319190, A320324, A321698, A321699, A322554, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322704 Number of regular hypergraphs on n labeled vertices with no singletons.

Original entry on oeis.org

1, 1, 2, 4, 80, 209944

Offset: 0

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is regular if all vertices have the same degree.

			The a(3) = 4 edge-sets:
  {}
  {{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A058891, A059441, A295193, A306021, A319189, A319190, A319612, A319729, A322698.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2,n}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2^n-n-1}],{n,1,5}]

A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

Original entry on oeis.org

1, 15, 33, 51, 55, 85, 91, 93, 123, 155, 161, 165, 177, 187, 201, 203, 205, 249, 255, 295, 299, 301, 327, 329, 335, 341, 377, 381, 415, 451, 465, 471, 511, 527, 537, 545, 553, 559, 561, 573, 611, 615, 633, 635, 649, 667, 679, 697, 703, 707, 723, 737, 785, 831

Offset: 1

Gus Wiseman, Sep 30 2019

nonn

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 terms together with their prime indices begins:
    1: {}
   15: {2,3}
   33: {2,5}
   51: {2,7}
   55: {3,5}
   85: {3,7}
   91: {4,6}
   93: {2,11}
  123: {2,13}
  155: {3,11}
  161: {4,9}
  165: {2,3,5}
  177: {2,17}
  187: {5,7}
  201: {2,19}
  203: {4,10}
  205: {3,13}
  249: {2,23}
  255: {2,3,7}
  295: {3,17}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The case including primes and nonsquarefree numbers is A320324.
The version for sum of prime indices is A327901.
The version for mean of prime indices is A327902.
Cf. A001222, A038041, A056239, A078175, A112798, A306017, A306021, A316413, A317583, A322794, A327908.

Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

A327901 Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).

Original entry on oeis.org

1, 35, 143, 209, 247, 391, 493, 629, 667, 851, 901, 1073, 1219, 1333, 1457, 1537, 1891, 1961, 2021, 2201, 2623, 2717, 2759, 2867, 2993, 3053, 3239, 3337, 3827, 3977, 4061, 4183, 4223, 4331, 4387, 4633, 5429, 5633, 5767, 5959, 6157, 6191, 6319, 7081, 7093, 7519

Offset: 1

Gus Wiseman, Sep 30 2019

nonn

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 terms together with their prime indices begins:
     1: {}
    35: {3,4}
   143: {5,6}
   209: {5,8}
   247: {6,8}
   391: {7,9}
   493: {7,10}
   629: {7,12}
   667: {9,10}
   851: {9,12}
   901: {7,16}
  1073: {10,12}
  1219: {9,16}
  1333: {11,14}
  1457: {11,15}
  1537: {10,16}
  1891: {11,18}
  1961: {12,16}
  2021: {14,15}
  2201: {11,20}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version including primes and nonsquarefree numbers is A326534.
The version for number of prime indices is A327900.
The version for mean of prime indices is A327902.
Cf. A001222, A005117, A056239, A112798, A302242, A306021, A320324, A326566, A326574, A327908.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Total/@primeMS/@primeMS[#]&];

A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 472, 23342

Offset: 0

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

			The a(3) = 2 hypergraphs:
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 5 hypergraphs:
  {{1},{2},{3},{4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The a(5) = 26 hypergraphs:
  {{1},{2},{3},{4},{5}}
  {{1,2},{1,3},{2,4},{3,5},{4,5}}
  {{1,2},{1,3},{2,5},{3,4},{4,5}}
  {{1,2},{1,4},{2,3},{3,5},{4,5}}
  {{1,2},{1,4},{2,5},{3,4},{3,5}}
  {{1,2},{1,5},{2,3},{3,4},{4,5}}
  {{1,2},{1,5},{2,4},{3,4},{3,5}}
  {{1,3},{1,4},{2,3},{2,5},{4,5}}
  {{1,3},{1,4},{2,4},{2,5},{3,5}}
  {{1,3},{1,5},{2,3},{2,4},{4,5}}
  {{1,3},{1,5},{2,4},{2,5},{3,4}}
  {{1,4},{1,5},{2,3},{2,4},{3,5}}
  {{1,4},{1,5},{2,3},{2,5},{3,4}}
  {{1,2,3},{1,2,4},{1,3,5},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,4},{1,4,5},{2,3,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,3,4},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,4,5},{2,3,4},{3,4,5}}
  {{1,2,3},{1,3,4},{1,4,5},{2,3,5},{2,4,5}}
  {{1,2,3},{1,3,5},{1,4,5},{2,3,4},{2,4,5}}
  {{1,2,4},{1,2,5},{1,3,4},{2,3,5},{3,4,5}}
  {{1,2,4},{1,2,5},{1,3,5},{2,3,4},{3,4,5}}
  {{1,2,4},{1,3,4},{1,3,5},{2,3,5},{2,4,5}}
  {{1,2,4},{1,3,5},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,5},{1,3,4},{1,3,5},{2,3,4},{2,4,5}}
  {{1,2,5},{1,3,4},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,3,4},{1,2,3,5},{1,2,4,5},{1,3,4,5},{2,3,4,5}}

Row sums of A322706.

Cf. A005176, A058891, A059441, A116539, A295193, A299353, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

A302129 Number of unlabeled uniform connected hypergraphs of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 6, 1, 9, 10, 17, 1, 108, 1, 86, 401, 482, 1, 4469, 1, 8435, 47959, 8082, 1, 1007342, 52414, 112835, 15338453, 11899367, 1, 362657533, 1, 977129970, 9349593479, 35787684, 1771297657, 390347162497, 1, 779945988, 9360467497257, 16838238535445

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

A hypergraph is uniform if all edges have the same size. The weight of a hypergraph is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(8) = 9 uniform connected hypergraphs:
  {{1,2,3,4,5,6,7,8}}
  {{1,2,3,7}, {4,5,6,7}}
  {{1,2,5,6}, {3,4,5,6}}
  {{1,3,4,5}, {2,3,4,5}}
  {{1,2}, {1,3}, {2,4}, {3,4}}
  {{1,3}, {2,4}, {3,5}, {4,5}}
  {{1,4}, {2,3}, {2,4}, {3,4}}
  {{1,4}, {2,5}, {3,5}, {4,5}}
  {{1,5}, {2,5}, {3,5}, {4,5}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000005, A007716, A038041, A038041, A048143, A299353, A301481, A301920, A301924, A306019, A306021, A331508.

\\ See A331508 for T(n, k).
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
a(n) = {if(n==0, 1, sumdiv(n, d, if(d==1 || d==n, d==1, InvEulerT(vector(d, i, T(n/d, i)))[d] )))} \\ Andrew Howroyd, Jan 16 2024

a(p) = 1 for prime p. - Andrew Howroyd, Jan 16 2024

a(11) onwards from Andrew Howroyd, Jan 16 2024

A320606 Regular triangle read by rows where T(n,k) is the number of k-uniform hypergraphs spanning n labeled vertices where every two vertices appear together in some edge, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 5, 1, 0, 0, 1, 388, 16, 1, 0, 0, 1, 477965, 27626, 42, 1

Offset: 1

Gus Wiseman, Jan 10 2019

			Triangle begins:
      1
      0      1
      0      0      1
      0      0      1      1
      0      0      1      5      1
      0      0      1    388     16      1
      0      0      1 477965  27626     42      1

Row sums are A321134. Column k = 3 is A302394 without the initial terms.

Cf. A058891, A116539, A301922, A306021, A319189, A320444.

Table[Length[Select[Subsets[If[k==0,{},Subsets[Range[n],{k}]]],And[Union@@#==Range[n],Length[Union@@(Subsets[#,{2}]&/@#)]==Binomial[n,2]]&]],{n,0,6},{k,0,n}]

A321134 Number of uniform hypergraphs spanning n vertices where every two vertices appear together in some edge.

Original entry on oeis.org

1, 1, 1, 2, 7, 406, 505635

Offset: 0

Gus Wiseman, Jan 10 2019

A hypergraph is uniform if all edges have the same size.

			The a(4) = 7 hypergraphs:
  {{1,2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Row sums of A320606.

Cf. A058891, A299353, A302394, A306021, A319189, A320444.

Table[Sum[Length[Select[Subsets[Subsets[Range[n],{k}]],And[Union@@#==Range[n],Length[Union@@(Subsets[#,{2}]&/@#)]==Binomial[n,2]]&]],{k,1,n}],{n,1,6}]

A322706 Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 12, 12, 1, 0, 1, 70, 330, 70, 1, 0, 1, 465, 11205, 11205, 465, 1, 0, 1, 3507, 505505, 2531200, 505505, 3507, 1, 0

Offset: 1

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

			Triangle begins:
  1
  1       0
  1       1       0
  1       3       1       0
  1      12      12       1       0
  1      70     330      70       1       0
  1     465   11205   11205     465       1       0
  1    3507  505505 2531200  505505    3507       1       0
Row 4 counts the following hypergraphs:
  {{1}{2}{3}{4}}  {{12}{13}{24}{34}}  {{123}{124}{134}{234}}
                  {{12}{14}{23}{34}}
                  {{13}{14}{23}{24}}

Row sums are A322705. Second column is A001205. Third column is A110101.

Cf. A005176, A058891, A059441, A295193, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Table[Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

cp's OEIS Frontend

A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

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A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.

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A322704 Number of regular hypergraphs on n labeled vertices with no singletons.

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A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

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A327901 Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).

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A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

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A302129 Number of unlabeled uniform connected hypergraphs of weight n.

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A320606 Regular triangle read by rows where T(n,k) is the number of k-uniform hypergraphs spanning n labeled vertices where every two vertices appear together in some edge, n >= 0, 0 <= k <= n.

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A321134 Number of uniform hypergraphs spanning n vertices where every two vertices appear together in some edge.

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