A320456 -id:A320456

A320461

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

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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}}.

LINKS

Table of n, a(n) for n=1..43.

Eric Weisstein's World of Mathematics, Simple Graph

EXAMPLE

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}}

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[#])]&]

CROSSREFS

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320456, A320458, A320459, A320462, A320532.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 13 2018

STATUS

approved

A329559

MM-numbers of multiset clutters (connected weak antichains of multisets).

+10
20

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A weak antichain of multisets is a multiset of multisets, none of which is a proper subset of any other.

LINKS

Table of n, a(n) for n=1..60.

FORMULA

Equals {1} followed by the intersection of A305078 and A316476.

EXAMPLE

The sequence of terms tother with their corresponding clutters begins:

1: {} 37: {{1,1,2}} 91: {{1,1},{1,2}}

2: {{}} 41: {{6}} 97: {{3,3}}

3: {{1}} 43: {{1,4}} 101: {{1,6}}

5: {{2}} 47: {{2,3}} 103: {{2,2,2}}

7: {{1,1}} 49: {{1,1},{1,1}} 107: {{1,1,4}}

9: {{1},{1}} 53: {{1,1,1,1}} 109: {{10}}

11: {{3}} 59: {{7}} 113: {{1,2,3}}

13: {{1,2}} 61: {{1,2,2}} 121: {{3},{3}}

17: {{4}} 67: {{8}} 125: {{2},{2},{2}}

19: {{1,1,1}} 71: {{1,1,3}} 127: {{11}}

23: {{2,2}} 73: {{2,4}} 131: {{1,1,1,1,1}}

25: {{2},{2}} 79: {{1,5}} 137: {{2,5}}

27: {{1},{1},{1}} 81: {{1},{1},{1},{1}} 139: {{1,7}}

29: {{1,3}} 83: {{9}} 149: {{3,4}}

31: {{5}} 89: {{1,1,1,2}} 151: {{1,1,2,2}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

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[100], And[stableQ[primeMS[#], Divisible], Length[zsm[primeMS[#]]]<=1]&]

CROSSREFS

Connected numbers are A305078.

Stable numbers are A316476.

Clutters (of sets) are A048143.

Cf. A056239, A112798, A289509, A302242, A302494, A304716, A318991, A319837, A320275, A320456, A328514, A329553, A329555.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 18 2019

STATUS

approved

A330060

MM-numbers of VDD-normalized multisets of multisets.

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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}}.

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

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}

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[#]]&]

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).

KEYWORD

nonn,eigen

AUTHOR

Gus Wiseman, Dec 03 2019

STATUS

approved

A330097

MM-numbers of VDD-normalized multiset partitions.

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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}}.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

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.

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[#]]&]

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).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 04 2019

STATUS

approved

A330108

MM-numbers of MM-normalized multisets of multisets.

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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}}

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

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}

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[#]]&]

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).

KEYWORD

nonn,eigen

AUTHOR

Gus Wiseman, Dec 05 2019

STATUS

approved

A330120

MM-numbers of lexicographically normalized multisets of multisets.

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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}}

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

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).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 05 2019

STATUS

approved

A330121

MM-numbers of lexicographically normalized multiset partitions.

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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}}

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

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).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 05 2019

STATUS

approved

A330122

MM-numbers of MM-normalized multiset partitions.

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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}}.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

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}

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[#]]&]

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).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 05 2019

STATUS

approved

A330123

BII-numbers of MM-normalized set-systems.

+10
19

0, 1, 3, 4, 5, 7, 11, 13, 15, 20, 21, 23, 31, 33, 37, 45, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 77, 79, 84, 85, 87, 95, 97, 101, 109, 116, 117, 119, 127, 139, 143, 159, 173, 180, 181, 183, 191, 195, 196, 197, 199, 203, 205, 207, 212, 213, 215, 223, 225, 229

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

First differs from A330110 in lacking 141 and having 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.

A set-system is a finite set of finite nonempty set of positive integers.

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 binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

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}}.

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

The sequence of all MM-normalized set-systems together with their BII-numbers begins:

0: {} 45: {{1},{3},{1,2},{2,3}}

1: {{1}} 52: {{1,2},{1,3},{2,3}}

3: {{1},{2}} 53: {{1},{1,2},{1,3},{2,3}}

4: {{1,2}} 55: {{1},{2},{1,2},{1,3},{2,3}}

5: {{1},{1,2}} 63: {{1},{2},{3},{1,2},{1,3},{2,3}}

7: {{1},{2},{1,2}} 64: {{1,2,3}}

11: {{1},{2},{3}} 65: {{1},{1,2,3}}

13: {{1},{3},{1,2}} 67: {{1},{2},{1,2,3}}

15: {{1},{2},{3},{1,2}} 68: {{1,2},{1,2,3}}

20: {{1,2},{1,3}} 69: {{1},{1,2},{1,2,3}}

21: {{1},{1,2},{1,3}} 71: {{1},{2},{1,2},{1,2,3}}

23: {{1},{2},{1,2},{1,3}} 75: {{1},{2},{3},{1,2,3}}

31: {{1},{2},{3},{1,2},{1,3}} 77: {{1},{3},{1,2},{1,2,3}}

33: {{1},{2,3}} 79: {{1},{2},{3},{1,2},{1,2,3}}

37: {{1},{1,2},{2,3}} 84: {{1,2},{1,3},{1,2,3}}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

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[0, 100], Sort[bpe/@bpe[#]]==mmnorm[bpe/@bpe[#]]&]

CROSSREFS

A subset of A326754.

Non-isomorphic multiset partitions are A007716.

Unlabeled spanning set-systems counted by vertices are A055621.

Unlabeled set-systems counted by weight are A283877.

MM-weight is A302242.

Cf. A000120, A000612, A048793, A056239, A070939, A112798, A300913, A319559, A320456, A326031, A330101, A330102, 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).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 05 2019

STATUS

approved

A339113

Products of primes of squarefree semiprime index (A322551).

+10
19

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 169, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 841, 907, 929, 937

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also MM-numbers of labeled multigraphs (without uncovered vertices). 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}}.

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

The sequence of terms together with the corresponding multigraphs begins:

1: {} 233: {{2,7}} 487: {{2,11}}

13: {{1,2}} 257: {{3,5}} 491: {{1,15}}

29: {{1,3}} 269: {{2,8}} 499: {{3,8}}

43: {{1,4}} 271: {{1,10}} 559: {{1,2},{1,4}}

47: {{2,3}} 293: {{1,11}} 577: {{1,16}}

73: {{2,4}} 313: {{3,6}} 607: {{2,12}}

79: {{1,5}} 347: {{2,9}} 611: {{1,2},{2,3}}

101: {{1,6}} 373: {{1,12}} 631: {{3,9}}

137: {{2,5}} 377: {{1,2},{1,3}} 647: {{1,17}}

139: {{1,7}} 389: {{4,5}} 653: {{4,7}}

149: {{3,4}} 421: {{1,13}} 673: {{1,18}}

163: {{1,8}} 439: {{3,7}} 677: {{2,13}}

167: {{2,6}} 443: {{1,14}} 727: {{2,14}}

169: {{1,2},{1,2}} 449: {{2,10}} 751: {{4,8}}

199: {{1,9}} 467: {{4,6}} 757: {{1,19}}

MATHEMATICA

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;

Select[Range[1000], FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; !sqfsemiQ[PrimePi[p]]]&]

CROSSREFS

These primes (of squarefree semiprime index) are listed by A322551.

The strict (squarefree) case is A309356.

The prime instead of squarefree semiprime version:

primes: A006450

products: A076610

strict: A302590

The nonprime instead of squarefree semiprime version:

primes: A007821

products: A320628

odd: A320629

strict: A340104

odd strict: A340105

The semiprime instead of squarefree semiprime version:

primes: A106349

products: A339112

strict: A340020

A001358 lists semiprimes, with odd/even terms A046315/A100484.

A002100 counts partitions into squarefree semiprimes.

A005117 lists squarefree numbers.

A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.

A056239 gives the sum of prime indices, which are listed by A112798.

A302242 is the weight of the multiset of multisets with MM-number n.

A305079 is the number of connected components for MM-number n.

A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).

A338899/A270650/A270652 give the prime indices of squarefree semiprimes.

A339561 lists products of distinct squarefree semiprimes (ranking: A339560).

MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).

Cf. A000040, A000720, A001055, A001222, A003963, A289509, A320461.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 12 2021

STATUS

approved