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 1-10 of 14 results. Next

A330060 MM-numbers of VDD-normalized multisets of multisets.

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, 63, 64, 70, 72, 74, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128
Offset: 1

Views

Author

Gus Wiseman, Dec 03 2019

Keywords

Comments

First differs from A330104 and A330120 in having 35 and lacking 69, with corresponding multisets of multisets 35: {{2},{1,1}} and 69: {{1},{2,2}}.
First differs from A330108 in having 207 and lacking 175, with corresponding multisets of multisets 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of all VDD-normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        49: {11}{11}         84: {}{}{1}{11}
   2: {}          24: {}{}{}{1}      52: {}{}{12}         89: {1112}
   3: {1}         26: {}{12}         53: {1111}           90: {}{1}{1}{2}
   4: {}{}        27: {1}{1}{1}      54: {}{1}{1}{1}      91: {11}{12}
   6: {}{1}       28: {}{}{11}       56: {}{}{}{11}       95: {2}{111}
   7: {11}        30: {}{1}{2}       57: {1}{111}         96: {}{}{}{}{}{1}
   8: {}{}{}      32: {}{}{}{}{}     60: {}{}{1}{2}       98: {}{11}{11}
   9: {1}{1}      35: {2}{11}        63: {1}{1}{11}      104: {}{}{}{12}
  12: {}{}{1}     36: {}{}{1}{1}     64: {}{}{}{}{}{}    105: {1}{2}{11}
  13: {12}        37: {112}          70: {}{2}{11}       106: {}{1111}
  14: {}{11}      38: {}{111}        72: {}{}{}{1}{1}    108: {}{}{1}{1}{1}
  15: {1}{2}      39: {1}{12}        74: {}{112}         111: {1}{112}
  16: {}{}{}{}    42: {}{1}{11}      76: {}{}{111}       112: {}{}{}{}{11}
  18: {}{1}{1}    45: {1}{1}{2}      78: {}{1}{12}       113: {123}
  19: {111}       48: {}{}{}{}{1}    81: {1}{1}{1}{1}    114: {}{1}{111}

Crossrefs

Equals the image/fixed points of the idempotent sequence A330061.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Select[Range[100],Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]

A330097 MM-numbers of VDD-normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 35, 37, 39, 45, 49, 53, 57, 63, 81, 89, 91, 95, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 245, 247, 259, 265, 267, 273, 281, 285, 311, 315, 329, 333, 339, 343
Offset: 1

Views

Author

Gus Wiseman, Dec 04 2019

Keywords

Comments

First differs from A330122 in having 207 and lacking 175, with corresponding multiset partitions 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
A multiset partition is a finite multiset of finite nonempty multisets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of all VDD-normalized multiset partitions together with their MM-numbers begins:
   1: 0             57: {1}{111}        151: {1122}
   3: {1}           63: {1}{1}{11}      159: {1}{1111}
   7: {11}          81: {1}{1}{1}{1}    161: {11}{22}
   9: {1}{1}        89: {1112}          165: {1}{2}{3}
  13: {12}          91: {11}{12}        169: {12}{12}
  15: {1}{2}        95: {2}{111}        171: {1}{1}{111}
  19: {111}        105: {1}{2}{11}      183: {1}{122}
  21: {1}{11}      111: {1}{112}        189: {1}{1}{1}{11}
  27: {1}{1}{1}    113: {123}           195: {1}{2}{12}
  35: {2}{11}      117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         245: {2}{11}{11}
  53: {1111}       147: {1}{11}{11}     247: {12}{111}
For example, 1155 is the MM-number of {{1},{2},{3},{1,1}}, which is VDD-normalized, so 1155 belongs to the sequence.
On the other hand, 69  is the MM-number of {{1},{2,2}}, but the VDD-normalization is {{2},{1,1}}, so 69 does not belong to the sequence.

Crossrefs

Equals the odd terms of A330060.
A subset of A320634.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330061, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]

A330104 MM-numbers of brute-force normalized multisets of multisets.

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, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 69, 72, 74, 76, 78, 81, 84, 89, 90, 91, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128, 131, 133
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2019

Keywords

Comments

First differs from A330060 and A330108 in having 69 and lacking 35, with corresponding multisets of multisets 69: {{1},{2,2}} and 35: {{2},{1,1}}.
First differs from A330120 in having 435 and lacking 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of all brute-force normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        52: {}{}{12}         89: {1112}
   2: {}          24: {}{}{}{1}      53: {1111}           90: {}{1}{1}{2}
   3: {1}         26: {}{12}         54: {}{1}{1}{1}      91: {11}{12}
   4: {}{}        27: {1}{1}{1}      56: {}{}{}{11}       96: {}{}{}{}{}{1}
   6: {}{1}       28: {}{}{11}       57: {1}{111}         98: {}{11}{11}
   7: {11}        30: {}{1}{2}       60: {}{}{1}{2}      104: {}{}{}{12}
   8: {}{}{}      32: {}{}{}{}{}     63: {1}{1}{11}      105: {1}{2}{11}
   9: {1}{1}      36: {}{}{1}{1}     64: {}{}{}{}{}{}    106: {}{1111}
  12: {}{}{1}     37: {112}          69: {1}{22}         108: {}{}{1}{1}{1}
  13: {12}        38: {}{111}        72: {}{}{}{1}{1}    111: {1}{112}
  14: {}{11}      39: {1}{12}        74: {}{112}         112: {}{}{}{}{11}
  15: {1}{2}      42: {}{1}{11}      76: {}{}{111}       113: {123}
  16: {}{}{}{}    45: {1}{1}{2}      78: {}{1}{12}       114: {}{1}{111}
  18: {}{1}{1}    48: {}{}{}{}{1}    81: {1}{1}{1}{1}    117: {1}{1}{12}
  19: {111}       49: {11}{11}       84: {}{}{1}{11}     120: {}{}{}{1}{2}

Crossrefs

Equals the image/fixed points of the idempotent sequence A330105.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A316983, A317533, A330098, A330101, A330103.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[100],Sort[primeMS/@primeMS[#]]==brute[primeMS/@primeMS[#]]&]

A330107 MM-numbers of brute-force normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 37, 39, 45, 49, 53, 57, 63, 69, 81, 89, 91, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 247, 259, 267, 273, 281, 285, 309, 311, 315, 329, 333, 339, 343, 351, 359
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2019

Keywords

Comments

A multiset partition is a finite multiset of finite nonempty multisets of positive integers.
We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of all brute-force normalized multiset partitions together with their MM-numbers begins:
   1: 0             63: {1}{1}{11}      159: {1}{1111}
   3: {1}           69: {1}{22}         161: {11}{22}
   7: {11}          81: {1}{1}{1}{1}    165: {1}{2}{3}
   9: {1}{1}        89: {1112}          169: {12}{12}
  13: {12}          91: {11}{12}        171: {1}{1}{111}
  15: {1}{2}       105: {1}{2}{11}      183: {1}{122}
  19: {111}        111: {1}{112}        189: {1}{1}{1}{11}
  21: {1}{11}      113: {123}           195: {1}{2}{12}
  27: {1}{1}{1}    117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         247: {12}{111}
  53: {1111}       147: {1}{11}{11}     259: {11}{112}
  57: {1}{111}     151: {1122}          267: {1}{1112}

Crossrefs

Equals the odd terms of A330104.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A300300, A316983, A317533, A320664, A330061, A330098, A330101, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==brute[primeMS/@primeMS[#]]&]

A330108 MM-numbers of MM-normalized multisets of multisets.

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, 63, 64, 70, 72, 74, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Comments

First differs from A330060 in having 175 and lacking 207, with corresponding multisets of multisets 175: {{2},{2},{1,1}} and 207: {{1},{1},{2,2}}.
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.
We define the MM-normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest MM-number.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}

Examples

			The sequence of all MM-normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        49: {11}{11}         84: {}{}{1}{11}
   2: {}          24: {}{}{}{1}      52: {}{}{12}         89: {1112}
   3: {1}         26: {}{12}         53: {1111}           90: {}{1}{1}{2}
   4: {}{}        27: {1}{1}{1}      54: {}{1}{1}{1}      91: {11}{12}
   6: {}{1}       28: {}{}{11}       56: {}{}{}{11}       95: {2}{111}
   7: {11}        30: {}{1}{2}       57: {1}{111}         96: {}{}{}{}{}{1}
   8: {}{}{}      32: {}{}{}{}{}     60: {}{}{1}{2}       98: {}{11}{11}
   9: {1}{1}      35: {2}{11}        63: {1}{1}{11}      104: {}{}{}{12}
  12: {}{}{1}     36: {}{}{1}{1}     64: {}{}{}{}{}{}    105: {1}{2}{11}
  13: {12}        37: {112}          70: {}{2}{11}       106: {}{1111}
  14: {}{11}      38: {}{111}        72: {}{}{}{1}{1}    108: {}{}{1}{1}{1}
  15: {1}{2}      39: {1}{12}        74: {}{112}         111: {1}{112}
  16: {}{}{}{}    42: {}{1}{11}      76: {}{}{111}       112: {}{}{}{}{11}
  18: {}{1}{1}    45: {1}{1}{2}      78: {}{1}{12}       113: {123}
  19: {111}       48: {}{}{}{}{1}    81: {1}{1}{1}{1}    114: {}{1}{111}

Crossrefs

Equals the image/fixed points of the idempotent sequence A330194.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[100],Sort[primeMS/@primeMS[#]]==mmnorm[primeMS/@primeMS[#]]&]

A330120 MM-numbers of lexicographically normalized multisets of multisets.

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, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 69, 72, 74, 76, 78, 81, 84, 89, 90, 91, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128, 131, 133
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Comments

First differs from A330104 in lacking 435 and having 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the lexicographic normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the lexicographically least of these representatives.
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}

Examples

			The sequence of all lexicographically normalized multisets of multisets together with their MM-numbers begins:
   1: 0          21: {1}{11}       52: {}{}{12}        89: {1112}
   2: {}         24: {}{}{}{1}     53: {1111}          90: {}{1}{1}{2}
   3: {1}        26: {}{12}        54: {}{1}{1}{1}     91: {11}{12}
   4: {}{}       27: {1}{1}{1}     56: {}{}{}{11}      96: {}{}{}{}{}{1}
   6: {}{1}      28: {}{}{11}      57: {1}{111}        98: {}{11}{11}
   7: {11}       30: {}{1}{2}      60: {}{}{1}{2}     104: {}{}{}{12}
   8: {}{}{}     32: {}{}{}{}{}    63: {1}{1}{11}     105: {1}{2}{11}
   9: {1}{1}     36: {}{}{1}{1}    64: {}{}{}{}{}{}   106: {}{1111}
  12: {}{}{1}    37: {112}         69: {1}{22}        108: {}{}{1}{1}{1}
  13: {12}       38: {}{111}       72: {}{}{}{1}{1}   111: {1}{112}
  14: {}{11}     39: {1}{12}       74: {}{112}        112: {}{}{}{}{11}
  15: {1}{2}     42: {}{1}{11}     76: {}{}{111}      113: {123}
  16: {}{}{}{}   45: {1}{1}{2}     78: {}{1}{12}      114: {}{1}{111}
  18: {}{1}{1}   48: {}{}{}{}{1}   81: {1}{1}{1}{1}   117: {1}{1}{12}
  19: {111}      49: {11}{11}      84: {}{}{1}{11}    120: {}{}{}{1}{2}

Crossrefs

A subset of A320456.
MM-weight is A302242.
Non-isomorphic multiset partitions are A007716.
Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105, A330194.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

A330121 MM-numbers of lexicographically normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 37, 39, 45, 49, 53, 57, 63, 69, 81, 89, 91, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 247, 259, 267, 273, 281, 285, 309, 311, 315, 329, 333, 339, 343, 351, 359
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Comments

First differs from A330107 in lacking 435 and having 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the lexicographic normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the lexicographically least of these representatives.
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}

Examples

			The sequence of all lexicographically normalized multiset partitions together with their MM-numbers begins:
   1:               63: {1}{1}{11}      159: {1}{1111}
   3: {1}           69: {1}{22}         161: {11}{22}
   7: {11}          81: {1}{1}{1}{1}    165: {1}{2}{3}
   9: {1}{1}        89: {1112}          169: {12}{12}
  13: {12}          91: {11}{12}        171: {1}{1}{111}
  15: {1}{2}       105: {1}{2}{11}      183: {1}{122}
  19: {111}        111: {1}{112}        189: {1}{1}{1}{11}
  21: {1}{11}      113: {123}           195: {1}{2}{12}
  27: {1}{1}{1}    117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         247: {12}{111}
  53: {1111}       147: {1}{11}{11}     259: {11}{112}
  57: {1}{111}     151: {1122}          267: {1}{1112}

Crossrefs

Equals the odd terms of A330120.
A subset of A320634.
MM-weight is A302242.
Non-isomorphic multiset partitions are A007716.
Cf. A056239, A112798, A317533, A320456, A330061, A330098, A330103, A330105, A330194.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

A330122 MM-numbers of MM-normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 35, 37, 39, 45, 49, 53, 57, 63, 81, 89, 91, 95, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 175, 183, 189, 195, 223, 225, 243, 245, 247, 259, 265, 267, 273, 281, 285, 311, 315, 329, 333, 339, 343
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Comments

We define the MM-normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest MM-number.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of all MM-normalized multiset partitions together with their MM-numbers begins:
   1: 0             57: {1}{111}        151: {1122}
   3: {1}           63: {1}{1}{11}      159: {1}{1111}
   7: {11}          81: {1}{1}{1}{1}    161: {11}{22}
   9: {1}{1}        89: {1112}          165: {1}{2}{3}
  13: {12}          91: {11}{12}        169: {12}{12}
  15: {1}{2}        95: {2}{111}        171: {1}{1}{111}
  19: {111}        105: {1}{2}{11}      175: {2}{2}{11}
  21: {1}{11}      111: {1}{112}        183: {1}{122}
  27: {1}{1}{1}    113: {123}           189: {1}{1}{1}{11}
  35: {2}{11}      117: {1}{1}{12}      195: {1}{2}{12}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         245: {2}{11}{11}
  53: {1111}       147: {1}{11}{11}     247: {12}{111}

Crossrefs

Equals the odd terms of A330108.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105, A330194.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==mmnorm[primeMS/@primeMS[#]]&]

A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 7, 8, 9, 6, 3, 12, 13, 14, 15, 16, 3, 18, 19, 12, 21, 6, 7, 24, 9, 26, 27, 28, 13, 30, 3, 32, 15, 6, 35, 36, 37, 38, 39, 24, 3, 42, 13, 12, 45, 14, 13, 48, 49, 18, 15, 52, 53, 54, 15, 56, 57, 26, 3, 60, 37, 6, 63, 64, 39, 30, 3, 12, 35, 70
Offset: 1

Views

Author

Gus Wiseman, Dec 03 2019

Keywords

Comments

We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of multisets is first by length and then lexicographically.
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}

Links

Crossrefs

This sequence is idempotent and its image/fixed points are A330060.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Map[Times@@Prime/@#&,sysnorm[primeMS/@primeMS[n]],{0,1}],{n,100}]

A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 2, 5, 11, 12, 13, 2, 7, 15, 16, 17, 18, 19, 4, 5, 3, 7, 2, 11, 23, 24, 25, 2, 13, 27, 4, 7, 29, 30, 31, 32, 3, 11, 2, 17, 35, 36, 37, 2, 19, 3, 13, 8, 5, 41, 6, 7, 43, 4, 11, 45, 2, 23, 47, 48, 49, 2, 25, 3, 17, 4, 13, 53, 54, 5, 11, 8
Offset: 1

Views

Author

Gus Wiseman, Aug 28 2022

Keywords

Comments

Row-products are the positive integers 1, 2, 3, ...

Examples

			The first 16 rows:
   1 =
   2 = 2
   3 = 3
   4 = 4
   5 = 5
   6 = 6
   7 = 7
   8 = 8
   9 = 9
  10 = 2 * 5
  11 = 11
  12 = 12
  13 = 13
  14 = 2 * 7
  15 = 15
  16 = 16
The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.

Crossrefs

Row-lengths are A287170, firsts A066205, even bisection A356229.
Applying bigomega to all parts gives A356226, statistics A356227-A356232.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@#&/@Split[primeMS[n],#1>=#2-1&],{n,100}]
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.