A303546 -id:A303546 - cp's OEIS frontend

A320813 Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.

1, 0, 1, 2, 5, 13, 33, 104, 293, 938, 2892

Offset: 0

Gus Wiseman, Nov 08 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 row sums are all > 1, (2) the positive entries in each row are relatively prime, and (3) 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(2) = 1 through a(5) = 13 multiset partitions:
  {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
                      {{1,2,3,4}}    {{1,2,2,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,3,3}}
                      {{1,3},{2,3}}  {{1,2,3,4,4}}
                                     {{1,2,3,4,5}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}

This is the case of A320804 where the underlying multiset is aperiodic.

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813, A321283, A321390.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
aperQ[m_]:=Length[m]==0||GCD@@Length/@Split[Sort[m]]==1;
Table[Length[Union[brute /@ Select[mpm[n],And[Min@@Length/@#>1,aperQ[Join@@#]&&And@@aperQ /@ #]&]]],{n,0,7}] (* Gus Wiseman, Jan 19 2024 *)

Definition corrected by Gus Wiseman, Jan 19 2024

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 0, 2, 0, 3, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 0, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 0, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 0, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

The positions of zeros in this sequence are the prime powers A000961.

			The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, A303710.

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[Select[facsr[n],GCD@@Length/@Split[#]===1&]],{n,100}]

a(n) = Sum_{d in A007916, d|A052409(n)} mu(d) * A303707(n^(1/d)).

A303547 Number of non-isomorphic periodic multiset partitions of weight n.

Original entry on oeis.org

0, 1, 1, 4, 1, 13, 1, 33, 10, 94, 1, 327, 1, 913, 100, 3017, 1, 10233, 1, 34236, 919, 119372, 1, 432234, 91, 1574227, 9945, 5916177, 1, 22734231, 1, 89003059, 119378, 356058543, 1000, 1453509039, 1, 6044132797, 1574233, 25612601420, 1, 110509543144, 1, 485161348076

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is periodic if its multiplicities have a common divisor greater than 1. For this sequence neither the parts nor their multiset union are required to be periodic, only the multiset of parts.

			Non-isomorphic representatives of the a(4) = 4 multiset partitions are {{1,1},{1,1}}, {{1,2},{1,2}}, {{1},{1},{1},{1}}, {{1},{1},{2},{2}}.

Cf. A000740, A000837, A001055, A007716, A007916, A049311, A100953, A255906, A269134, A275024, A293993, A301700, A303386, A303431, A303546.

a(n) = 1 if n is prime.

a(n) = A007716(n) - A303546(n).

More terms from Jinyuan Wang, Jun 21 2020

A321283 Number of non-isomorphic multiset partitions of weight n in which the part sizes are relatively prime.

Original entry on oeis.org

1, 1, 2, 7, 21, 84, 214, 895, 2607, 9591, 31134, 119313, 400950, 1574123, 5706112, 22572991, 86933012, 356058243, 1427784135, 6044132304, 25342935667, 110414556330, 481712291885, 2166488898387, 9784077216457, 45369658599779, 211869746691055, 1011161497851296, 4871413403219085

Offset: 0

Gus Wiseman, Nov 06 2018

nonn

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 the row sums are relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is aperiodic, where 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) = 21 multiset partitions with relatively prime part-sizes:
  {{1}}  {{1},{1}}  {{1},{1,1}}    {{1},{1,1,1}}
         {{1},{2}}  {{1},{2,2}}    {{1},{1,2,2}}
                    {{1},{2,3}}    {{1},{2,2,2}}
                    {{2},{1,2}}    {{1},{2,3,3}}
                    {{1},{1},{1}}  {{1},{2,3,4}}
                    {{1},{2},{2}}  {{2},{1,2,2}}
                    {{1},{2},{3}}  {{3},{1,2,3}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{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}}
Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with aperiodic multiset union:
  {{1}}  {{1,2}}    {{1,2,2}}      {{1,2,2,2}}
         {{1},{2}}  {{1,2,3}}      {{1,2,3,3}}
                    {{1},{2,2}}    {{1,2,3,4}}
                    {{1},{2,3}}    {{1},{2,2,2}}
                    {{2},{1,2}}    {{1,2},{2,2}}
                    {{1},{2},{2}}  {{1},{2,3,3}}
                    {{1},{2},{3}}  {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

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

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A320800-A320810.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), 1 + sum(d=1, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A,i*d)*x^(i*d)))) )))} \\ Andrew Howroyd, Jan 17 2023

\\ faster self contained program.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=vector(n, t, K(q, t, n\t))); s+=permcount(q)*polcoef(sum(d=1, n, moebius(d)*exp(sum(t=1, n\d, sum(i=1, n\(t*d), u[t][i*d]*x^(i*d*t))/t, O(x*x^n)) )), n)); s/n!)} \\ Andrew Howroyd, Jan 17 2023

a(n) = A007716(n) - A320810(n). - Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

Original entry on oeis.org

0, 2, 3, 12, 7, 84, 15, 410, 354, 3073, 56, 28300, 101, 210036, 126839, 2070047, 297, 25295952, 490, 269662769, 89071291, 3449056162, 1255, 51132696310, 400625539, 713071048480, 145126661415, 11351097702297, 4565, 199926713003444, 6842, 3460838122540969

Offset: 1

Gus Wiseman, Nov 15 2018

nonn

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 the column sums are not relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.
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(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,1},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,2}}
                      {{1,2},{2,2}}
                      {{1,2},{3,3}}
                      {{1,2},{3,4}}
                      {{1,3},{2,3}}
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
  {{1},{1}}  {{1},{1,1}}    {{1,1,2,2}}        {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1},{1,1,1}}      {{1,1},{1,1,1}}
                            {{1,1},{1,1}}      {{1},{1},{1,1,1}}
                            {{1},{1,2,2}}      {{1},{1,1},{1,1}}
                            {{1,1},{2,2}}      {{1},{1},{1},{1,1}}
                            {{1,2},{1,2}}      {{1},{1},{1},{1},{1}}
                            {{1},{1},{1,1}}
                            {{1},{1},{2,2}}
                            {{1},{2},{1,2}}
                            {{1},{1},{1},{1}}
                            {{1},{1},{2},{2}}

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

Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n));Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A,i*d)*x^(i*d)))) ))), -n)} \\ Andrew Howroyd, Jan 17 2023

a(n) = A007716(n) - A321283(n). - Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A300275 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} 1/(1 - x^n)^n.

Original entry on oeis.org

1, 2, 5, 10, 23, 40, 85, 147, 276, 474, 858, 1421, 2484, 4079, 6850, 11137, 18333, 29277, 47329, 74768, 118703, 185614, 290782, 449568, 696009, 1066258, 1632376, 2479057, 3759611, 5661568, 8512308, 12722132, 18974109, 28157619, 41690937, 61453929, 90379783

Offset: 1

Ilya Gutkovskiy, Mar 01 2018

nonn

Moebius transform of A000219.
From Gus Wiseman, Jan 21 2019: (Start)
Also the number of plane partitions of n with relatively prime entries. For example, the a(4) = 10 plane partitions are:
  31   211   1111
.
  3   21   11   111
  1   1    11   1
.
  2   11
  1   1
  1   1
.
  1
  1
  1
  1
Also the number of plane partitions of n whose multiset of rows is aperiodic, meaning its multiplicities are relatively prime. For example, the a(4) = 10 plane partitions are:
  4   31   22   211   1111
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000219, A000837, A078374, A300274, A300276, A300277, A300278.

Cf. A100953, A303546, A320802, A323584, A323585, A323587.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*sigma[2](j), j=1..n)/n)
    end:
a:= n-> add(b(d)*mobius(n/d), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018

nn = 37; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[1/(1 - x^n)^n, {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[1/(1 - x^k)^k, {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 37}]

a(n) = Sum_{d|n} mu(n/d)*A000219(d).

A320800 Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.

Original entry on oeis.org

1, 1, 1, 5, 14, 78, 157, 881, 2267, 9257, 28397

Offset: 0

Gus Wiseman, Nov 02 2018

The latter condition is equivalent to the parts having relatively prime sizes.
A multiset is aperiodic if its multiplicities are relatively prime.
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}}.
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) = 14 multiset partitions:
  {{1}}  {{1},{2}}  {{1},{2,2}}    {{1},{2,2,2}}
                    {{1},{2,3}}    {{1},{2,3,3}}
                    {{2},{1,2}}    {{1},{2,3,4}}
                    {{1},{2},{2}}  {{2},{1,2,2}}
                    {{1},{2},{3}}  {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{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, A303431, A303546, A303547, A316983, A320801-A320810.

A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Gus Wiseman, Aug 06 2018

			Triangle begins:
   1:  0
   2:  0  1
   3:  0  1
   4:  0  1  1
   5:  0  1
   6:  1  0  0  1
   7:  0  1
   8:  0  2  0  1
   9:  0  1  1
  10:  1  0  0  1
  11:  0  1
  12:  2  1  0  0  0  1
  13:  0  1
  14:  1  0  0  1
  15:  1  0  0  1
  16:  0  3  1  0  1
  17:  0  1
  18:  2  0  1  0  0  1
  19:  0  1
  20:  2  1  0  0  0  1

Row lengths are A000005. Row sums are A001055. First column is A281116. Number of nonzero terms in each row is A317751.
Cf. A007562, A007716, A014963, A027750, A045778, A050370, A162247, A289509, A303386, A303546.
Cf. A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[goc[n,d],{n,30},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A326026 Number of non-isomorphic multiset partitions of weight n where each part has a different length.

Original entry on oeis.org

1, 1, 2, 7, 12, 35, 111, 247, 624, 1843, 6717, 15020, 46847, 124808, 412577, 1658973, 4217546, 12997734, 40786810, 126971940, 437063393, 2106317043, 5499108365, 19037901867, 59939925812, 210338815573, 683526043801, 2741350650705, 14848209030691, 41533835240731, 151548411269815

Offset: 0

Gus Wiseman, Jul 13 2019

nonn

The number of non-isomorphic multiset partitions of weight n is A007716(n).

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{1},{1,1}}  {{1,2,3,3}}
                  {{1},{2,2}}  {{1,2,3,4}}
                  {{1},{2,3}}  {{1},{1,1,1}}
                  {{2},{1,2}}  {{1},{1,2,2}}
                               {{1},{2,2,2}}
                               {{1},{2,3,3}}
                               {{1},{2,3,4}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

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

Row sums of A332260.

Cf. A007716, A007837, A303546, A306017, A316980, A316983, A319560, A326514, A326517, A326533.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p,n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); polcoef(prod(k=1, #u, 1 + u[k]*x^k + O(x*x^n)), n)/prod(i=1, #v, i^v[i]*v[i]!)}
a(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Feb 08 2020

Terms a(11) and beyond from Andrew Howroyd, Feb 08 2020

A320804 Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.

Original entry on oeis.org

1, 0, 1, 2, 6, 13, 41, 104, 326, 958, 3096, 9958, 33869, 116806, 417741, 1526499, 5732931, 22015642, 86543717, 347495480, 1424832602, 5959123908, 25407212843, 110344848622, 487879651220, 2194697288628, 10039367091586, 46675057440634, 220447539120814

Offset: 0

Gus Wiseman, Nov 06 2018

nonn

Also the number of nonnegative integer matrices with (1) sum of elements equal to n, (2) no zero columns, (3) no rows summing to 0 or 1, and (4) no rows whose nonzero entries have a common divisor > 1, up to row and column permutations.
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(2) = 1 through a(5) = 13 multiset partitions with aperiodic parts and no singletons:
  {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
                      {{1,2,3,4}}    {{1,2,2,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,4,4}}
                      {{1,3},{2,3}}  {{1,2,3,4,5}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}

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

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
S(q, t, k)={Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t))  + O(x*x^k), -k)}
a(n)={if(n==0, 1, my(mbt=vector(n, d, moebius(d)), s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(dirmul(mbt, sum(t=1, n, K(q, t, n)/t)) - sum(t=1, n, S(q, t, n)/t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

cp's OEIS Frontend

A320813 Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A303547 Number of non-isomorphic periodic multiset partitions of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A321283 Number of non-isomorphic multiset partitions of weight n in which the part sizes are relatively prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A300275 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} 1/(1 - x^n)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A320800 Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.

Views

Author

Keywords

Comments

Examples

Crossrefs

A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

Views

Author

Keywords

Examples

Crossrefs

Programs