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.

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A330105 MM-number of the brute-force 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, 69, 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, 69, 138
Offset: 1

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Author

Gus Wiseman, Dec 02 2019

Keywords

Comments

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}}.

Crossrefs

This sequence is idempotent and its image/fixed points are A330104.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A316983, A317533, A330098, A330103.
Other normalizations: A330061 (VDD MM), A330101 (brute-force BII), A330102 (VDD BII), A330105 (brute-force MM).
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
    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]}];
Table[Map[Times@@Prime/@#&,brute[primeMS/@primeMS[n]],{0,1}],{n,100}]

A330230 Least MM-number of a multiset of multisets with n distinct representatives obtainable by permuting the vertices.

Original entry on oeis.org

1, 35, 141, 1713, 28011, 355
Offset: 1

Views

Author

Gus Wiseman, Dec 09 2019

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 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 terms together with their corresponding multisets of multisets begins:
      1: {}
     35: {{2},{1,1}}
    141: {{1},{2,3}}
   1713: {{1},{2,3,4}}
  28011: {{1},{2,3,4,5}}
    355: {{2},{1,1,3}}

Crossrefs

The BII-number version is A330218.
Positions of first appearances in A330098.
The sorted version is A330233.
MM-numbers of achiral multisets of multisets are A330232.
MM-numbers of fully-chiral multisets of multisets are A330236.
Cf. A001055, A003238, A007716, A056239, A112798, A302242, A303975, A322847, A330103, A330223, A330227, A330231, A330236.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]];
dv=Table[Length[graprms[primeMS/@primeMS[n]]],{n,1000}];
Table[Position[dv,i][[1,1]],{i,First[Split[Union[dv],#1+1==#2&]]}]

A330233 Least MM-numbers of multisets of multisets with a given number of distinct representatives (obtainable by vertex-permutations).

Original entry on oeis.org

1, 35, 141, 1713, 28011, 355, 34567, 4045, 54849, 64615, 15265, 95363, 126841
Offset: 1

Views

Author

Gus Wiseman, Dec 09 2019

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 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 terms together with their corresponding multisets of multisets begins:
       1: {}
      35: {{2},{1,1}}
     141: {{1},{2,3}}
     355: {{2},{1,1,3}}
    1713: {{1},{2,3,4}}
    4045: {{2},{1,1,3,4}}
   15265: {{2},{1,4},{1,1,3}}
   28011: {{1},{2,3,4,5}}
   34567: {{1,2},{3,4,5}}
   54849: {{1},{2,3},{4,5}}
   64615: {{2},{1,1,3,4,5}}
   95363: {{2,3},{1,1,4,5}}
  126841: {{3},{1,2},{1,4,5}}

Crossrefs

Sorted positions of first appearances in A330098.
The unsorted version is A330230.
The BII-number version is A330218.
MM-numbers of achiral multisets of multisets are A330232.
MM-numbers of fully-chiral multisets of multisets are A330236.
Cf. A001055, A003238, A007716, A056239, A112798, A302242, A303975, A322847, A330103, A330223, A330227, A330231.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]];
dv=Table[Length[graprms[primeMS/@primeMS[n]]],{n,1000}];
Table[Position[dv,i][[1,1]],{i,First/@Gather[dv]}]

A330281 Numbers whose prime-indices do not have weakly increasing numbers of distinct prime factors.

Original entry on oeis.org

221, 247, 299, 403, 442, 494, 533, 598, 663, 689, 741, 767, 806, 871, 884, 897, 899, 988, 1066, 1079, 1105, 1189, 1196, 1209, 1235, 1261, 1326, 1339, 1378, 1417, 1482, 1495, 1517, 1534, 1537, 1547, 1599, 1612, 1651, 1703, 1711, 1729, 1742, 1768, 1794, 1798
Offset: 1

Views

Author

Gus Wiseman, Dec 10 2019

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.

Examples

			The sequence of terms together with their prime indices begins:
   221: {6,7}
   247: {6,8}
   299: {6,9}
   403: {6,11}
   442: {1,6,7}
   494: {1,6,8}
   533: {6,13}
   598: {1,6,9}
   663: {2,6,7}
   689: {6,16}
   741: {2,6,8}
   767: {6,17}
   806: {1,6,11}
   871: {6,19}
   884: {1,1,6,7}
For example, 884 has prime indices {1,1,6,7} with numbers of distinct prime factors (0,0,2,1), which is not weakly increasing, so 884 belongs to the sequence.

Links

Crossrefs

The version where prime factors are counted with multiplicity is A330103.
Cf. A001221, A001222, A056239, A112798, A302242, A330098, A330230, A330233.

Programs

  • Mathematica
    Select[Range[1000],!OrderedQ[PrimeNu/@PrimePi/@First/@FactorInteger[#]]&]
Previous Showing 11-14 of 14 results.

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