A303546 -id:A303546 - cp's OEIS frontend

A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

1, 1, 1, 4, 8, 22, 34, 84, 137, 271, 450, 857, 1373, 2483, 3993, 6823, 10990, 18332, 28966, 47328, 74286, 118614, 184755, 290781, 448010, 695986, 1063773, 1632100, 2474970, 3759610, 5654233, 8512307, 12710995, 18973247, 28139285, 41690830, 61423271, 90379782

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.

			The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
  4   31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
  31   211   1111
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323585, A323587.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.

A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.

Original entry on oeis.org

1, 1, 0, 3, 7, 21, 30, 83, 129, 267, 428, 856, 1332, 2482, 3909, 6798, 10853, 18331, 28665, 47327, 73829, 118527, 183898, 290780, 446508, 695964, 1061290, 1631829, 2470970, 3759609, 5646952, 8512306, 12700005, 18972387, 28120953, 41690725, 61392966, 90379781

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

			The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:
  31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The same for a(5) = 21:
  41   32   311   221   2111
.
  4   3   31   21   22   21   211   111   1111
  1   2   1    2    1    11   1     11    1
.
  3   2   21   11   111
  1   2   1    11   1
  1   1   1    1    1
.
  2   11
  1   1
  1   1
  1   1

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323584, A323587.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,GCD@@Length/@Split[Transpose[PadRight[#]]]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the second is A323584.

A323587 Number of strict (distinct parts) plane partitions of n with relatively prime parts.

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 6, 10, 12, 18, 30, 40, 48, 74, 92, 142, 172, 242, 294, 412, 490, 722, 854, 1164, 1396, 1880, 2260, 2976, 3748, 4764, 5792, 7472, 9082, 11488, 14012, 17522, 21830, 26896, 32820, 40536, 49488, 60636, 73626, 89962, 108854, 134240, 160952, 195858

Offset: 0

Gus Wiseman, Jan 20 2019

nonn

			The a(9) = 18 plane partitions:
  81   72   621   54   531   432
.
  8   7   61   62   5   51   53   42   43
  1   2   2    1    4   3    1    3    2
.
  6   5   4
  2   3   3
  1   1   2

Cf. A000219, A000837, A003293, A006951, A026007, A100883, A117433 (strict plane partitions), A300275 (plane partitions with relatively prime parts), A303546, A320802, A323584, A323585.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]}],{n,30}]

Moebius transform of A117433.

A304450 Numbers that are not perfect powers and whose prime factors span an initial interval of prime numbers.

Original entry on oeis.org

2, 6, 12, 18, 24, 30, 48, 54, 60, 72, 90, 96, 108, 120, 150, 162, 180, 192, 210, 240, 270, 288, 300, 360, 384, 420, 432, 450, 480, 486, 540, 600, 630, 648, 720, 750, 768, 810, 840, 864, 960, 972, 1050, 1080, 1152, 1200, 1260, 1350, 1440, 1458, 1470, 1500, 1536

Offset: 1

Gus Wiseman, May 12 2018

nonn

The multiset of prime indices of a(n) is the a(n)-th row of A112798. This multiset is normal, meaning it spans an initial interval of positive integers, and aperiodic, meaning its multiplicities are relatively prime.

			Sequence of all normal aperiodic multisets begins
2:   {1}
6:   {1,2}
12:  {1,1,2}
18:  {1,2,2}
24:  {1,1,1,2}
30:  {1,2,3}
48:  {1,1,1,1,2}
54:  {1,2,2,2}
60:  {1,1,2,3}
72:  {1,1,1,2,2}
90:  {1,2,2,3}
96:  {1,1,1,1,1,2}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
150: {1,2,3,3}
162: {1,2,2,2,2}
180: {1,1,2,2,3}
192: {1,1,1,1,1,1,2}
210: {1,2,3,4}
240: {1,1,1,1,2,3}
270: {1,2,2,2,3}
288: {1,1,1,1,1,2,2}
300: {1,1,2,3,3}
360: {1,1,1,2,2,3}
384: {1,1,1,1,1,1,1,2}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000740, A000837, A000961, A001597, A001694, A005117, A007916, A052409, A052410, A055932, A056239, A112798, A303431, A303546, A303945.

Select[Range[1000],FactorInteger[#][[-1,1]]==Prime[Length[FactorInteger[#]]]&&GCD@@FactorInteger[#][[All,2]]===1&]

ok(n)={my(f=factor(n)[,1]); #f && !ispower(n) && #f==primepi(f[#f])} \\ Andrew Howroyd, Aug 26 2018

Intersection of A007916 and A055932.

A320805 Number of non-isomorphic multiset partitions of weight n in which each part, as well as the multiset union of the parts, is an aperiodic multiset.

Original entry on oeis.org

1, 1, 2, 6, 16, 55, 139, 516, 1500, 5269, 17017

Offset: 0

Gus Wiseman, Nov 07 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the positive entries in each row are relatively prime and (2) the column sums are relatively prime.
A multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 16 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,2}}      {{1,2,2,2}}
         {{1},{2}}  {{1,2,3}}      {{1,2,3,3}}
                    {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A303710, A320800-A320810, A321283.

A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 11, 8, 1, 10, 21, 27, 16, 1, 12, 38, 61, 63, 32, 1, 18, 57, 120, 162, 143, 64, 1, 22, 87, 205, 347, 409, 319, 128, 1, 28, 122, 333, 651, 950, 1000, 703, 256, 1, 32, 164, 506, 1132, 1926, 2504, 2391, 1535, 512, 1, 42, 217, 734, 1840

Offset: 1

Gus Wiseman, May 15 2018

A multiset is normal if it spans an initial interval of positive integers, and is aperiodic if its multiplicities are relatively prime.

			Triangle begins:
1
1    2
1    4    4
1    6   11    8
1   10   21   27   16
1   12   38   61   63   32
1   18   57  120  162  143   64
1   22   87  205  347  409  319  128
The a(4,3) = 11 multisets are (3), (13), (23), (113), (123), (133), (223), (233), (1123), (1223), (1233).

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A303976.

Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945, A303974.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Max],{n,10}]

T(n,k) = sum(j=1, n, sumdiv(j, d, sum(i=max(1, j+k-n), d, moebius(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1)))) \\ Andrew Howroyd, Jan 20 2023

T(n,k) = Sum_{j=1..n} Sum_{d|j} Sum_{i=max(1, j+k-n)..d} mu(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1). - Andrew Howroyd, Jan 20 2023

A320807 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic and all parts of the dual are also aperiodic.

Original entry on oeis.org

1, 1, 3, 6, 17, 41, 122, 345, 1077, 3385, 11214

Offset: 0

Gus Wiseman, Nov 07 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of entries equal to n and no zero rows or columns, in which each row and each column has relatively prime nonzero entries.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{2}}  {{2},{1,2}}    {{1},{2,3,4}}
                    {{1},{1},{1}}  {{1,2},{3,4}}
                    {{1},{2},{2}}  {{1,3},{2,3}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810.

cp's OEIS Frontend

A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

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A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.

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A323587 Number of strict (distinct parts) plane partitions of n with relatively prime parts.

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A304450 Numbers that are not perfect powers and whose prime factors span an initial interval of prime numbers.

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A320805 Number of non-isomorphic multiset partitions of weight n in which each part, as well as the multiset union of the parts, is an aperiodic multiset.

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A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

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A320807 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic and all parts of the dual are also aperiodic.

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