A321717 -id:A321717

A321719

Number of non-normal semi-magic squares with sum of entries equal to n.

+10
25

1, 1, 3, 7, 28, 121, 746, 5041, 40608, 362936, 3635017, 39916801, 479206146, 6227020801, 87187426839, 1307674521272, 20923334906117, 355687428096001, 6402415241245577, 121645100408832001, 2432905938909013343, 51090942176372298027, 1124001180562929946213

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

OFFSET

0,3

COMMENTS

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Squares must be of size k X k where k is a divisor of n. This implies that a(p) = p! + 1 for p prime since the only allowable squares are of sizes 1 X 1 and p X p. The 1 X 1 square is [p], the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 1 for n > 1. - Chai Wah Wu, Jan 13 2019

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..100

Wikipedia, Magic square

Index entries for sequences related to magic squares

FORMULA

a(p) = p! + 1 for p prime and a(n) >= n! + 1 for n > 1 (see comment above). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n} A257493(d, n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

EXAMPLE

The a(3) = 7 semi-magic squares:

[3]

.

[1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]

[0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]

[0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

MATHEMATICA

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];

multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];

Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#], SameQ@@Total/@prs2mat[#], SameQ@@Total/@Transpose[prs2mat[#]]]&]], {n, 5}]

CROSSREFS

Cf. A006052, A120732, A120733, A257493, A271103, A319056, A319616.

Cf. A321717, A321718, A321720, A321721, A321722, A321723, A321724.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

a(7) from Chai Wah Wu, Jan 13 2019

a(6) corrected and a(8)-a(15) added by Chai Wah Wu, Jan 14 2019

a(16)-a(19) from Chai Wah Wu, Jan 16 2019

Terms a(20) and beyond from Andrew Howroyd, Apr 11 2020

STATUS

approved

A319189

Number of uniform regular hypergraphs spanning n vertices.

+10
18

1, 1, 2, 3, 10, 29, 3780, 5012107

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

OFFSET

0,3

COMMENTS

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is uniform if all edges have the same size, and regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.

Also the number of 0-1 matrices with n columns, all distinct rows, no zero columns, equal row-sums, and equal column-sums, up to a permutation of the rows.

LINKS

Table of n, a(n) for n=0..7.

EXAMPLE

The a(4) = 10 edge-sets:

{{1,2},{3,4}}

{{1,3},{2,4}}

{{1,4},{2,3}}

{{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}}

{{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Inequivalent representatives of the a(4) = 10 matrices:

[1 1 1 1]

.

[1 1 0 0] [1 0 1 0] [1 0 0 1]

[0 0 1 1] [0 1 0 1] [0 1 1 0]

.

[1 0 0 0] [1 1 0 0] [1 1 0 0] [1 0 1 0] [1 1 1 0]

[0 1 0 0] [1 0 1 0] [1 0 0 1] [1 0 0 1] [1 1 0 1]

[0 0 1 0] [0 1 0 1] [0 1 1 0] [0 1 1 0] [1 0 1 1]

[0 0 0 1] [0 0 1 1] [0 0 1 1] [0 1 0 1] [0 1 1 1]

.

[1 1 0 0]

[1 0 1 0]

[1 0 0 1]

[0 1 1 0]

[0 1 0 1]

[0 0 1 1]

MATHEMATICA

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

CROSSREFS

Uniform hypergraphs are counted by A306021. Unlabeled uniform regular multiset partitions are counted by A319056. Regular graphs are A295193. Uniform clutters are A299353.

Cf. A002829, A005176, A007016, A049311, A058891, A101370, A110100, A110101, A321717, A321720.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 17 2018

EXTENSIONS

a(7) from Jinyuan Wang, Jun 20 2020

STATUS

approved

A321718

Number of coupled non-normal semi-magic rectangles with sum of entries equal to n.

+10
14

1, 1, 5, 9, 44, 123, 986, 5043, 45832, 366300, 3862429, 39916803, 495023832, 6227020803, 88549595295, 1308012377572, 21086922542349, 355687428096003, 6427700493998229, 121645100408832003, 2437658338007783347, 51091307195905020227, 1125098837523651728389, 25852016738884976640003, 620752163206546966698620, 15511210044577707492319496

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

OFFSET

0,3

COMMENTS

A coupled non-normal semi-magic rectangle is a nonnegative integer matrix with equal row sums and equal column sums. The common row sum may be different from the common column sum.

Rectangles must be of size k X m where k and m are divisors of n. This implies that a(p) = p! + 3 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. The 1 X 1 square is [p], the 1 X p and p X 1 rectangles are [1,...,1] and its transpose and the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..69

Wikipedia, Magic square

FORMULA

a(p) = p! + 3 for p prime. a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

EXAMPLE

The a(3) = 9 coupled semi-magic rectangles:

[3] [1 1 1]

.

[1] [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]

[1] [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]

[1] [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

MATHEMATICA

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];

multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];

Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], SameQ@@Total/@prs2mat[#], SameQ@@Total/@Transpose[prs2mat[#]]]&]], {n, 5}]

CROSSREFS

Cf. A006052, A120733, A271103, A319056, A321654.

Cf. A321717, A321719, A321720, A321721, A321722, A321724, A321724.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

a(7)-a(15) from Chai Wah Wu, Jan 15 2019

a(16)-a(19) from Chai Wah Wu, Jan 16 2019

Terms a(20) onward from Max Alekseyev, Dec 04 2024

STATUS

approved

A321721

Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.

+10
14

1, 1, 2, 2, 4, 2, 7, 2, 10, 7, 12, 2, 38, 2, 21, 46, 72, 2, 162, 2, 420, 415, 64, 2, 4987, 1858, 110, 9336, 45456, 2, 136018, 2, 1014658, 406578, 308, 3996977, 34937078, 2, 502, 28010167, 1530292965, 2, 508164038, 2, 54902992348, 51712929897, 1269, 2, 3217847072904, 8597641914, 9168720349613

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

OFFSET

0,3

COMMENTS

A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d, for some d|n.

LINKS

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

Wikipedia, Magic square

FORMULA

a(p) = 2 for p prime corresponding to the 1 X 1 square [p] and the permutation matrices of size p X p with partition (1...10...0). - Chai Wah Wu, Jan 16 2019

a(n) = Sum_{d|n} A333733(d,n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

EXAMPLE

Non-isomorphic representatives of the a(2) = 2 through a(6) = 7 multiset partitions:

{{1}{2}} {{1}{2}{3}} {{11}{22}} {{1}{2}{3}{4}{5}} {{111}{222}}

{{12}{12}} {{112}{122}}

{{1}{2}{3}{4}} {{11}{22}{33}}

{{11}{23}{23}}

{{12}{13}{23}}

{{1}{2}{3}{4}{5}{6}}

Inequivalent representatives of the a(6) = 7 matrices:

[6]

.

[3 0] [2 1]

[0 3] [1 2]

.

[2 0 0] [2 0 0] [1 1 0]

[0 2 0] [0 1 1] [1 0 1]

[0 0 2] [0 1 1] [0 1 1]

.

[1 0 0 0 0 0]

[0 1 0 0 0 0]

[0 0 1 0 0 0]

[0 0 0 1 0 0]

[0 0 0 0 1 0]

[0 0 0 0 0 1]

Inequivalent representatives of the a(9) = 7 matrices:

[9]

.

[3 0 0] [3 0 0] [2 1 0] [2 1 0] [1 1 1]

[0 3 0] [0 2 1] [1 1 1] [1 0 2] [1 1 1]

[0 0 3] [0 1 2] [0 1 2] [0 2 1] [1 1 1]

.

[1 0 0 0 0 0 0 0 0]

[0 1 0 0 0 0 0 0 0]

[0 0 1 0 0 0 0 0 0]

[0 0 0 1 0 0 0 0 0]

[0 0 0 0 1 0 0 0 0]

[0 0 0 0 0 1 0 0 0]

[0 0 0 0 0 0 1 0 0]

[0 0 0 0 0 0 0 1 0]

[0 0 0 0 0 0 0 0 1]

CROSSREFS

Cf. A006052, A007716, A057150, A120732, A271103, A319056, A319616.

Cf. A321717, A321718, A321719, A321722, A321724, A333733.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

a(11)-a(13) from Chai Wah Wu, Jan 16 2019

a(14)-a(15) from Chai Wah Wu, Jan 20 2019

Terms a(16) and beyond from Andrew Howroyd, Apr 11 2020

STATUS

approved

A326565

Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

+10
8

1, 0, 1, 1, 4, 13, 91, 1318, 73581, 51913025

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

OFFSET

0,5

COMMENTS

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

LINKS

Table of n, a(n) for n=0..9.

EXAMPLE

The a(2) = 1 through a(5) = 13 antichains:

{{1,4},{2,3}} {{1,2,5},{1,3,4}}

{{2,4},{1,2,3}} {{1,3,5},{2,3,4}}

{{3,4},{1,2,4}} {{1,4,5},{2,3,5}}

{{1,4,5},{1,2,3,4}}

{{2,3,5},{1,2,3,4}}

{{2,4,5},{1,2,3,5}}

{{3,4,5},{1,2,4,5}}

{{1,5},{2,4},{1,2,3}}

{{2,5},{3,4},{1,2,4}}

{{3,5},{1,2,5},{1,3,4}}

{{4,5},{1,3,5},{2,3,4}}

{{1,4,5},{2,3,5},{1,2,3,4}}

MATHEMATICA

stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];

cleq[n_]:=Select[stableSets[Subsets[Range[n], {2, n}], SubsetQ[#1, #2]||Total[#1]!=Total[#2]&], Union@@#==Range[n]&];

Table[Length[cleq[n]], {n, 0, 5}]

CROSSREFS

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326360, A326518, A326534, A326566.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 13 2019

EXTENSIONS

a(9) from Andrew Howroyd, Aug 14 2019

STATUS

approved

A321720

Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.

+10
7

1, 1, 2, 6, 25, 120, 726, 5040, 40410, 362881, 3630840, 39916800, 479069574, 6227020800, 87181402140, 1307674370040, 20922977418841, 355687428096000, 6402388104196400, 121645100408832000, 2432903379962038320, 51090942171778378800, 1124000886592995642000, 25852016738884976640000

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

OFFSET

0,3

COMMENTS

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..180

Wikipedia, Magic square

Index entries for sequences related to magic squares

FORMULA

a(p) = p! for p prime as the squares are all permutation matrices of order p and a(n) >= n! for n > 1 (see comments in A321717 and A321719). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n, d<=n/d} A008300(n/d, d) for n > 0. - Andrew Howroyd, Apr 11 2020

MATHEMATICA

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];

Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#], SameQ@@Total/@prs2mat[#], SameQ@@Total/@Transpose[prs2mat[#]]]&]], {n, 5}]

CROSSREFS

Cf. A006052, A007016, A057151, A068313, A008300, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321719, A321722, A321723.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

a(7) from Chai Wah Wu, Jan 13 2019

a(8)-a(15) from Chai Wah Wu, Jan 14 2019

a(16)-a(21) from Chai Wah Wu, Jan 16 2019

Terms a(22) and beyond from Andrew Howroyd, Apr 11 2020

STATUS

approved

A326566

Number of covering antichains of subsets of {1..n} with equal edge-sums.

+10
7

2, 1, 1, 2, 4, 14, 92, 1320, 73584, 51913039

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

OFFSET

0,1

COMMENTS

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

LINKS

Table of n, a(n) for n=0..9.

EXAMPLE

The a(1) = 1 through a(5) = 14 antichains:

{{3},{1,2}} {{1,4},{2,3}} {{1,2,5},{1,3,4}}

{{2,4},{1,2,3}} {{1,3,5},{2,3,4}}

{{3,4},{1,2,4}} {{1,4,5},{2,3,5}}

{{5},{1,4},{2,3}}

{{1,4,5},{1,2,3,4}}

{{2,3,5},{1,2,3,4}}

{{2,4,5},{1,2,3,5}}

{{3,4,5},{1,2,4,5}}

{{1,5},{2,4},{1,2,3}}

{{2,5},{3,4},{1,2,4}}

{{3,5},{1,2,5},{1,3,4}}

{{4,5},{1,3,5},{2,3,4}}

{{1,4,5},{2,3,5},{1,2,3,4}}

MATHEMATICA

stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];

cleq[n_]:=Select[stableSets[Subsets[Range[n]], SubsetQ[#1, #2]||Total[#1]!=Total[#2]&], Union@@#==Range[n]&];

Table[Length[cleq[n]], {n, 0, 5}]

CROSSREFS

The non-covering case is A326574.

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326518, A326534, A326565.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 13 2019

EXTENSIONS

a(9) from Andrew Howroyd, Aug 14 2019

STATUS

approved

A321723

Number of non-normal magic squares whose entries are all 0 or 1 and sum to n.

+10
6

1, 1, 0, 0, 9, 20, 96, 656, 5584, 48913, 494264, 5383552, 65103875, 840566080, 11834159652, 176621049784, 2838040416201, 48060623405312

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

OFFSET

0,5

COMMENTS

A non-normal magic square is a square matrix with row sums, column sums, and both diagonals all equal to d, for some d|n.

LINKS

Table of n, a(n) for n=0..17.

Wikipedia, Magic square

Index entries for sequences related to magic squares

FORMULA

a(n) >= A007016(n) with equality if n is prime. - Chai Wah Wu, Jan 15 2019

EXAMPLE

The a(4) = 9 magic squares:

[1 1]

.

[1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]

[0 0 1 0][0 0 0 1][0 0 1 0][0 0 0 1][1 0 0 0][0 1 0 0][1 0 0 0][0 1 0 0]

[0 0 0 1][0 1 0 0][1 0 0 0][0 0 1 0][0 1 0 0][0 0 0 1][0 0 1 0][1 0 0 0]

[0 1 0 0][0 0 1 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 1 0 0][0 0 1 0]

MATHEMATICA

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];

multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];

Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#], SameQ@@Join[{Tr[prs2mat[#]], Tr[Reverse[prs2mat[#]]]}, Total/@prs2mat[#], Total/@Transpose[prs2mat[#]]]]&]], {n, 5}]

CROSSREFS

Cf. A006052, A007016, A057151, A068313, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321718, A321719, A321720, A321721, A321722.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

a(7)-a(15) from Chai Wah Wu, Jan 15 2019

a(16)-a(17) from Chai Wah Wu, Jan 16 2019

STATUS

approved

A326574

Number of antichains of subsets of {1..n} with equal edge-sums.

+10
6

2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575

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

OFFSET

0,1

COMMENTS

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

LINKS

Table of n, a(n) for n=0..9.

EXAMPLE

The a(0) = 2 through a(4) = 22 antichains:

{} {} {} {} {}

{{}} {{}} {{}} {{}} {{}}

{{3},{1,2}} {{2,3}}

{{3},{1,2}}

{{4},{1,3}}

{{1,4},{2,3}}

{{2,4},{1,2,3}}

{{3,4},{1,2,4}}

MATHEMATICA

stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];

cleqset[set_]:=stableSets[Subsets[set], SubsetQ[#1, #2]||Total[#1]!=Total[#2]&];

Table[Length[cleqset[Range[n]]], {n, 0, 5}]

CROSSREFS

Set partitions with equal block-sums are A035470.

Antichains with different edge-sums are A326030.

MM-numbers of multiset partitions with equal part-sums are A326534.

The covering case is A326566.

Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 18 2019

EXTENSIONS

a(9) from Andrew Howroyd, Aug 13 2019

STATUS

approved

A321698

MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

+10
4

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 41, 43, 47, 49, 51, 53, 55, 59, 64, 67, 73, 79, 81, 83, 85, 93, 97, 101, 103, 109, 113, 121, 123, 125, 127, 128, 131, 137, 139, 149, 151, 155, 157, 161, 163, 165, 167, 169, 177, 179

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

OFFSET

1,2

COMMENTS

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 and regular, so its MM-number 15463 belongs to the sequence.

LINKS

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

EXAMPLE

The sequence of all uniform regular multiset multisystems, together with their MM-numbers, begins:

1: {} 33: {{1},{3}} 109: {{10}}

2: {{}} 41: {{6}} 113: {{1,2,3}}

3: {{1}} 43: {{1,4}} 121: {{3},{3}}

4: {{},{}} 47: {{2,3}} 123: {{1},{6}}

5: {{2}} 49: {{1,1},{1,1}} 125: {{2},{2},{2}}

7: {{1,1}} 51: {{1},{4}} 127: {{11}}

8: {{},{},{}} 53: {{1,1,1,1}} 128: {{},{},{},{},{},{}}

9: {{1},{1}} 55: {{2},{3}} 131: {{1,1,1,1,1}}

11: {{3}} 59: {{7}} 137: {{2,5}}

13: {{1,2}} 64: {{},{},{},{},{},{}} 139: {{1,7}}

15: {{1},{2}} 67: {{8}} 149: {{3,4}}

16: {{},{},{},{}} 73: {{2,4}} 151: {{1,1,2,2}}

17: {{4}} 79: {{1,5}} 155: {{2},{5}}

19: {{1,1,1}} 81: {{1},{1},{1},{1}} 157: {{12}}

23: {{2,2}} 83: {{9}} 161: {{1,1},{2,2}}

25: {{2},{2}} 85: {{2},{4}} 163: {{1,8}}

27: {{1},{1},{1}} 93: {{1},{5}} 165: {{1},{2},{3}}

29: {{1,3}} 97: {{3,3}} 167: {{2,6}}

31: {{5}} 101: {{1,6}} 169: {{1,2},{1,2}}

32: {{},{},{},{},{}} 103: {{2,2,2}} 177: {{1},{7}}

MATHEMATICA

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

Select[Range[100], And[SameQ@@PrimeOmega/@primeMS[#], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

CROSSREFS

Cf. A005176, A007016, A112798, A271103, A283877, A299353, A302242, A306017, A319056, A319189, A320324, A321699, A321717, A322554, A322703, A322833.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 27 2018

STATUS

approved