cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A320456 Numbers whose multiset multisystem spans an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117
Offset: 1

Views

Author

Gus Wiseman, Oct 13 2018

Keywords

Comments

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 n-th multiset multisystem 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 78th multiset multisystem is {{},{1},{1,2}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}

Crossrefs

Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.
Cf. A320458, A320459, A320461, A320462, A320463, A320464, A320532, A320533.

Programs

  • Mathematica
    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[100],normQ[primeMS/@primeMS[#]]&]

A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 91, 161, 299, 329, 377, 611, 667, 1261, 1363, 1937, 2021, 2093, 2117, 2639, 4277, 4669, 7567, 8671, 8827, 9541, 13559, 14053, 14147, 14819, 15617, 16211, 17719, 23989, 24017, 26273, 27521, 28681, 29003, 31349, 31913, 36569, 44551, 44603, 46483, 48691
Offset: 1

Views

Author

Gus Wiseman, Oct 13 2018

Keywords

Comments

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}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
     1: {}
     7: {{1,1}}
    13: {{1,2}}
    91: {{1,1},{1,2}}
   161: {{1,1},{2,2}}
   299: {{2,2},{1,2}}
   329: {{1,1},{2,3}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
   667: {{2,2},{1,3}}
  1261: {{3,3},{1,2}}
  1363: {{1,3},{2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2093: {{1,1},{2,2},{1,2}}
  2117: {{1,3},{2,4}}
  2639: {{1,1},{1,2},{1,3}}
  4277: {{1,1},{1,2},{2,3}}
  4669: {{1,1},{2,2},{1,3}}

Links

Crossrefs

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.
Cf. A320456, A320458, A320459, A320462, A320532.

Programs

  • Mathematica
    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[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

A320533 MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 19, 37, 49, 53, 61, 89, 91, 113, 131, 133, 151, 161, 169, 223, 247, 251, 259, 281, 299, 311, 329, 343, 359, 361, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 637, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 931, 953
Offset: 1

Views

Author

Gus Wiseman, Oct 14 2018

Keywords

Comments

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}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
    1: {}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  133: {{1,1},{1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  259: {{1,1},{1,1,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}
  311: {{1,1,1,1,1,1}}
  329: {{1,1},{2,3}}
  343: {{1,1},{1,1},{1,1}}

Links

Crossrefs

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.
Cf. A320456, A320462, A320463, A320464, A320532.

Programs

  • Mathematica
    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[#]],And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

A320275 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 7, 9, 13, 19, 27, 37, 49, 53, 61, 81, 89, 91, 113, 131, 151, 169, 223, 243, 247, 251, 281, 299, 311, 343, 359, 361, 377, 427, 463, 503, 593, 611, 637, 659, 689, 703, 719, 729, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1183, 1291, 1321, 1339
Offset: 1

Views

Author

Gus Wiseman, Dec 16 2018

Keywords

Comments

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}}. This sequence lists all MM-numbers of not necessarily strict connected antichains of multisets spanning an initial interval of positive integers.

Examples

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
    2: {{}}
    3: {{1}}
    7: {{1,1}}
    9: {{1},{1}}
   13: {{1,2}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   37: {{1,1,2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   81: {{1},{1},{1},{1}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  169: {{1,2},{1,2}}

Crossrefs

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A286520, A290103, A293994, A302242, A316476, A319496, A319837, A320456, A320532.

Programs

  • Mathematica
    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]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]

A320463 MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 113, 377, 611, 1291, 1363, 1469, 1937, 2021, 2117, 3277, 4537, 4859, 5249, 5311, 7423, 8249, 8507, 16211, 16403, 16559, 16783, 16837, 17719, 20443, 20453, 24553, 25477, 26273, 26969, 27521, 34567, 37439, 39437, 41689, 42011, 42137, 42601, 43873, 43957
Offset: 1

Views

Author

Gus Wiseman, Oct 13 2018

Keywords

Comments

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}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    113: {{1,2,3}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1291: {{1,2,3,4}}
   1363: {{1,3},{2,3}}
   1469: {{1,2},{1,2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
   3277: {{1,3},{1,2,3}}
   4537: {{1,2},{1,3,4}}
   4859: {{1,4},{1,2,3}}
   5249: {{1,3},{1,2,4}}
   5311: {{2,3},{1,2,3}}
   7423: {{1,2},{2,3,4}}
   8249: {{2,4},{1,2,3}}
   8507: {{2,3},{1,2,4}}
  16211: {{1,2},{1,3},{1,4}}

Links

Crossrefs

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.
Cf. A320456, A320458, A320464, A320532, A320533.

Programs

  • Mathematica
    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[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]

A320464 MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 113, 169, 377, 611, 1291, 1363, 1469, 1937, 2021, 2117, 2197, 3277, 4537, 4859, 4901, 5249, 5311, 7423, 7943, 8249, 8507, 10933, 12769, 16211, 16403, 16559, 16783, 16837, 17719, 19097, 20443, 20453, 24553, 25181, 25477, 26273, 26969, 27521, 28561, 28717
Offset: 1

Views

Author

Gus Wiseman, Oct 13 2018

Keywords

Comments

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}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
     1: {}
    13: {{1,2}}
   113: {{1,2,3}}
   169: {{1,2},{1,2}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
  1291: {{1,2,3,4}}
  1363: {{1,3},{2,3}}
  1469: {{1,2},{1,2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2117: {{1,3},{2,4}}
  2197: {{1,2},{1,2},{1,2}}
  3277: {{1,3},{1,2,3}}
  4537: {{1,2},{1,3,4}}
  4859: {{1,4},{1,2,3}}
  4901: {{1,2},{1,2},{1,3}}
  5249: {{1,3},{1,2,4}}
  5311: {{2,3},{1,2,3}}
  7423: {{1,2},{2,3,4}}
  7943: {{1,2},{1,2},{2,3}}
  8249: {{2,4},{1,2,3}}
  8507: {{2,3},{1,2,4}}

Links

Crossrefs

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.
Cf. A320456, A320459, A320463, A320532, A320533.

Programs

  • Mathematica
    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[10000],And[normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]

A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 151, 223, 247, 251, 281, 299, 311, 359, 377, 427, 463, 503, 593, 611, 659, 689, 703, 719, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1291, 1321, 1339, 1363, 1511, 1619, 1703, 1733, 1739, 1757, 1769
Offset: 1

Views

Author

Gus Wiseman, Dec 16 2018

Keywords

Comments

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}}. This sequence lists all MM-numbers of connected strict antichains of multisets spanning an initial interval of positive integers.

Examples

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
    2: {{}}
    3: {{1}}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}

Crossrefs

Cf. A003963, A006126, A055932, A056239, A112798, A285573, A286520, A293994, A302242, A318401, A319719, A319837, A320275, A320456, A320532.

Programs

  • Mathematica
    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]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]

A320634 Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 35, 37, 39, 45, 49, 53, 57, 61, 63, 65, 69, 75, 81, 89, 91, 95, 105, 111, 113, 117, 131, 133, 135, 141, 143, 145, 147, 151, 159, 161, 165, 169, 171, 175, 183, 185, 189, 195, 207, 223, 225, 243, 245, 247, 251, 259, 265, 267, 273
Offset: 1

Views

Author

Gus Wiseman, Oct 18 2018

Keywords

Comments

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 n-th multiset multisystem 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 78th multiset multisystem is {{},{1},{1,2}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
    1: {}
    3: {{1}}
    7: {{1,1}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   19: {{1,1,1}}
   21: {{1},{1,1}}
   27: {{1},{1},{1}}
   35: {{2},{1,1}}
   37: {{1,1,2}}
   39: {{1},{1,2}}
   45: {{1},{1},{2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   57: {{1},{1,1,1}}
   61: {{1,2,2}}
   63: {{1},{1},{1,1}}
   65: {{2},{1,2}}
   69: {{1},{2,2}}
   75: {{1},{2},{2}}
   81: {{1},{1},{1},{1}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  111: {{1},{1,1,2}}
  113: {{1,2,3}}
  117: {{1},{1},{1,2}}
  131: {{1,1,1,1,1}}
  133: {{1,1},{1,1,1}}
  135: {{1},{1},{1},{2}}

Crossrefs

Odd terms of A320456.
Cf. A003963, A055932, A056239, A112798, A255906, A302242, A305052, A320532, A320629.

Programs

  • Mathematica
    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[1,100,2],normQ[primeMS/@primeMS[#]]&]
Showing 1-8 of 8 results.

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