A255397 -id:A255397 - cp's OEIS frontend

A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 1, 2, 1, 2, 2, 3, 2, 2, 2, 1, 0, 2, 1, 3, 2, 3, 3, 3, 1, 1, 3, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 4, 3, 2, 2, 4, 2, 1, 2, 3, 1, 4, 0, 3, 2, 1, 1, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 1, 4, 1, 1, 3, 2, 2, 3, 1, 4

Offset: 1

Gus Wiseman, Apr 03 2018

nonn

A multiset multisystem is a finite multiset of finite multisets of positive integers. The n-th multiset multisystem is constructed by factoring n into prime numbers and then factoring each prime index into prime numbers and taking their prime indices. This produces a unique multiset multisystem for each n, and every possible multiset multisystem is so constructed as n ranges over all positive integers.

			Sequence of finite multisets of finite multisets of positive integers begins: (), (()), ((1)), (()()), ((2)), (()(1)), ((11)), (()()()), ((1)(1)), (()(2)), ((3)), (()()(1)), ((12)), (()(11)), ((1)(2)), (()()()()), ((4)), (()(1)(1)), ((111)), (()()(2)).

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A001222, A003963, A007716, A034691, A056239, A061775, A063834, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302243.

with(numtheory):
a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 07 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[PrimeOmega/@primeMS[n]],{n,100}]

a(n,f=factor(n))=sum(i=1,#f~, bigomega(primepi(f[i,1]))*f[i,2]) \\ Charles R Greathouse IV, Nov 10 2021

A317533 Regular triangle read rows: T(n,k) = number of non-isomorphic multiset partitions of size n and length k.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 14, 9, 5, 7, 28, 33, 16, 7, 11, 69, 104, 74, 29, 11, 15, 134, 294, 263, 142, 47, 15, 22, 285, 801, 948, 599, 263, 77, 22, 30, 536, 2081, 3058, 2425, 1214, 453, 118, 30, 42, 1050, 5212, 9769, 9276, 5552, 2322, 761, 181, 42, 56, 1918, 12645, 29538, 34172, 23770, 11545, 4179, 1223, 267, 56

Offset: 1

Gus Wiseman, Jul 30 2018

			Non-isomorphic representatives of the T(3,2) = 4 multiset partitions:
  {{1},{1,1}}
  {{1},{1,2}}
  {{1},{2,2}}
  {{1},{2,3}}
Triangle begins:
    1
    2    2
    3    4    3
    5   14    9    5
    7   28   33   16    7
   11   69  104   74   29   11
   15  134  294  263  142   47   15

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

Row sums are A007716. First and last columns are both A000041.

Cf. A034691, A255397, A255903, A255906, A317532, A334550.

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
T[n_, k_] := M[k, n, n] - M[k - 1, n, n];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 08 2020, after Andrew Howroyd *)

\\ See A318795 for definition of M.
T(n,k)={M(k, n, n) - M(k-1, n, n)}
for(n=1, 10, for(k=1, n, print1(T(n,k),", "));print) \\ Andrew Howroyd, Dec 28 2019

\\ Faster version.
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, n)={1/prod(j=1, #q, (1-x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
G(m,n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}
A(n,m=n)={my(p=sum(k=0, m, G(k,n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}
{ my(T=A(10)); for(n=1, #T, print(T[n,1..n])) } \\ Andrew Howroyd, Aug 30 2020

Terms a(29) and beyond from Andrew Howroyd, Dec 28 2019

A302243 Total weight of the n-th twice-odd-factored multiset partition.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 1, 3, 3, 2, 2, 3, 2, 1, 2, 3, 3, 3, 1, 2, 3, 2, 4, 2, 4, 2, 4, 1, 3, 4, 3, 1, 3, 3, 2, 3, 3, 2, 4, 1, 2, 3, 4, 4, 2, 4, 2, 3, 2, 3, 4, 3, 1, 4, 3, 3, 4, 3, 2, 2, 3, 1, 3, 5, 5, 4, 2, 2, 3, 3, 3, 5, 2, 4, 3, 2, 1, 5, 4, 2, 3, 2, 4, 5, 4, 4

Offset: 0

Gus Wiseman, Apr 03 2018

nonn

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. The n-th twice-odd-factored multiset partition is constructed by factoring 2n + 1 into prime numbers and then factoring each prime index into prime numbers and taking their prime indices.

			Sequence of multiset partitions begins: (), ((1)), ((2)), ((11)), ((1)(1)), ((3)), ((12)), ((1)(2)), ((4)), ((111)), ((1)(11)), ((22)), ((2)(2)), ((1)(1)(1)), ((13)), ((5)), ((1)(3)), ((2)(11)), ((112)), ((1)(12)), ((6)).

Cf. A003963, A007716, A034691, A048673, A056239, A061775, A063834 A064216, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302242.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Sum[PrimeOmega[k],{k,primeMS[2n-1]}],{n,100}]

a(n) = A302242(2n + 1).

A317532 Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 34, 26, 8, 16, 124, 168, 76, 16, 32, 448, 962, 674, 208, 32, 64, 1568, 5224, 5344, 2392, 544, 64, 128, 5448, 27336, 39834, 24578, 7816, 1376, 128, 256, 18768, 139712, 283864, 236192, 99832, 24048, 3392, 256, 512, 64448, 702496, 1960320, 2161602, 1186866, 370976, 70656, 8192, 512

Offset: 1

Gus Wiseman, Jul 30 2018

			The T(3,2) = 8 multiset partitions:
  {{1},{1,1}}
  {{1},{2,2}}
  {{2},{1,2}}
  {{1},{1,2}}
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
Triangle begins:
    1
    2    2
    4    8    4
    8   34   26    8
   16  124  168   76   16
   32  448  962  674  208   32
  ...

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

Row sums are A255906.

Cf. A007716, A034691, A255397, A255903.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],Length[#]==k&]],{n,7},{k,n}]

\\ here B(n,k) is A239473(n,k).
B(n,k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
Row(n)={Vecrev(sum(j=1, n, B(n,j)*polcoef(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^binomial(k+j-1,j-1)), n))/y)}
{ for(n=1, 10, print(Row(n))) } \\ Andrew Howroyd, Dec 31 2019

Terms a(29) and beyond from Andrew Howroyd, Dec 31 2019

A339113 Products of primes of squarefree semiprime index (A322551).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

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}}.

			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}}

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.

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]

A339112 Products of primes of semiprime index (A106349).

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A semiprime (A001358) is a product of any two prime numbers.

Also MM-numbers of labeled multigraphs with loops (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}}.

			The sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):
     1:            149:   (34)     313:     (36)
     7:   (11)     161: (11)(22)   329:   (11)(23)
    13:   (12)     163:   (18)     343: (11)(11)(11)
    23:   (22)     167:   (26)     347:     (29)
    29:   (13)     169: (12)(12)   373:     (1C)
    43:   (14)     199:   (19)     377:   (12)(13)
    47:   (23)     203: (11)(13)   389:     (45)
    49: (11)(11)   227:   (44)     421:     (1D)
    73:   (24)     233:   (27)     439:     (37)
    79:   (15)     257:   (35)     443:     (1E)
    91: (11)(12)   269:   (28)     449:     (2A)
    97:   (33)     271:   (1A)     467:     (46)
   101:   (16)     293:   (1B)     487:     (2B)
   137:   (25)     299: (12)(22)   491:     (1F)
   139:   (17)     301: (11)(14)   499:     (38)

Robert Israel, Table of n, a(n) for n = 1..10000

These primes (of semiprime index) are listed by A106349.
The strict (squarefree) case is A340020.
The prime instead of semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The squarefree semiprime instead of semiprime version:
  strict: A309356
  primes: A322551
  products: A339113
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A056239 gives the sum of prime indices, which are listed by A112798.
A084126 and A084127 give the prime factors of semiprimes.
A101048 counts partitions into semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320892 lists even-omega non-products of distinct semiprimes.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A338898, A338912, and A338913 give the prime indices of semiprimes.
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, A005117, A007097, A289509, A320461.

N:= 1000: # for terms up to N
SP:= {}: p:= 1:
for i from 1 do
  p:= nextprime(p);
  if 2*p > N then break fi;
  Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
  SP:= SP union Q;
od:
SP:= sort(convert(SP,list)):
PSP:= map(ithprime,SP):
R:= {1}:
for p in PSP do
  Rp:= {}:
  for k from 1 while p^k <= N do
    Rpk:= select(`<=`,R, N/p^k);
    Rp:= Rp union map(`*`,Rpk, p^k);
  od;
  R:= R union Rp;
od:
sort(convert(R,list)); # Robert Israel, Nov 03 2024

semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]

A317546 Number of multimin partitions of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 18, 42, 104, 246, 594, 1416, 3391, 8084, 19312, 46041, 109829, 261827, 624254, 1487981, 3546883, 8453770, 20149014, 48021864, 114451536, 272769936, 650084053, 1549312743

Offset: 1

Gus Wiseman, Jul 31 2018

nonn

A multimin partition of m is an ordered multiset partition of m such that the minima of the blocks are weakly increasing.

			The a(3) = 7 multimin partitions of integer partitions of 3:
  (3),
  (1)(2), (12),
  (1)(1)(1), (1)(11), (11)(1), (111).
The a(4) = 18 multimin partitions of integer partitions of 4:
  (4),
  (1)(3), (13),
  (2)(2), (22),
  (1)(1)(2), (1)(12), (11)(2), (12)(1), (112),
  (1)(1)(1)(1), (1)(1)(11), (1)(11)(1), (1)(111), (11)(1)(1), (11)(11), (111)(1), (1111).

Cf. A007716, A020639, A034691, A255397, A300335, A317545.

mmcount[m_List]:=mmcount[m]=If[Length[m]===0,0,1+Plus@@mmcount/@Union[Subsets[Rest[m]]]];
Table[Sum[mmcount[Reverse[ptn]],{ptn,IntegerPartitions[n]}],{n,25}]

a(n) = Sum_{k > 0 : A056239(k) = n} A317545(k).

A317545 Number of multimin factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 4, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 5, 1, 16, 2, 2, 2, 11, 1, 2, 2, 12, 1, 5, 1, 5, 5, 2, 1, 28, 2, 4, 2, 5, 1, 8, 2, 12, 2, 2, 1, 15, 1, 2, 5, 32, 2, 5, 1, 5, 2, 5, 1, 29, 1, 2, 4, 5, 2, 5, 1, 28, 8, 2, 1, 15, 2, 2, 2, 12, 1, 12, 2, 5, 2, 2, 2, 64, 1, 4, 5, 11, 1, 5, 1, 12, 5

Offset: 1

Gus Wiseman, Jul 31 2018

nonn

A multimin factorizations of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing.

			The a(36) = 11 multimin factorizations:
  (36),
  (2*18), (4*9), (6*6), (12*3), (18*2),
  (2*2*9), (2*6*3), (4*3*3), (6*2*3),
  (2*2*3*3).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007716, A020639, A034691, A255397, A296560, A300335, A317546.

a[n_]:=If[n==1,1,Sum[a[d],{d,Divisors[n/FactorInteger[n][[1,1]]]}]];
Array[a,100]

A317545(n) = if(1==n,1,my(spf = factor(n)[1,1]); sumdiv(n/spf,d,A317545(d))); \\ Antti Karttunen, Sep 10 2018

memo317545 = Map(); \\ Memoized version.
A317545(n) = if(1==n,1,if(mapisdefined(memo317545, n), mapget(memo317545, n), my(spf = factor(n)[1,1], v = sumdiv(n/spf,d,A317545(d))); mapput(memo317545, n, v); (v))); \\ Antti Karttunen, Sep 10 2018

a(1) = 1; a(n > 1) = Sum_{d|(n/p)} a(d), where p is the smallest prime dividing n.

More terms from Antti Karttunen, Sep 10 2018

A317508 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with weakly decreasing sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 4, 1, 7, 2, 2, 2, 6, 1, 2, 2, 7, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 5, 2, 7, 2, 2, 1, 7, 1, 2, 4, 11, 2, 4, 1, 4, 2, 4, 1, 9, 1, 2, 3, 4, 2, 4, 1, 11, 5, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(60) = 7 split partitions:
  (3)(2)(1)(1)
  (32)(1)(1)
  (3)(21)(1)
  (3)(2)(11)
  (321)(1)
  (32)(11)
  (3211)

Cf. A001970, A056239, A063834, A255397, A296150, A316223, A317545, A317546, A319002, A319004.

Cf. A316245, A317715, A318434, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Length[Select[compositionPartitions[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],OrderedQ[Total/@#]&]],{n,100}]

A319001 Number of ordered multiset partitions of integer partitions of n where the sequence of GCDs of the partitions is weakly increasing.

Original entry on oeis.org

1, 1, 3, 7, 18, 42, 105, 248, 606, 1450, 3507, 8415, 20305, 48785, 117502, 282574, 680137, 1636005, 3936841, 9470776, 22787529, 54822530, 131901491, 317336519, 763489051, 1836862947, 4419324581, 10632404189, 25580507505, 61543948594, 148068421107

Offset: 0

Gus Wiseman, Sep 07 2018

nonn

If we form a multiorder by treating integer partitions (a,...,z) as multiarrows GCD(a, ..., z) <= {z, ..., a}, then a(n) is the number of triangles of weight n.

			The a(4) = 18 ordered multiset partitions:
  {{4}}   {{1,3}}    {{2,2}}     {{1,1,2}}       {{1,1,1,1}}
         {{1},{3}}  {{2},{2}}   {{1},{1,2}}     {{1},{1,1,1}}
                                {{1,2},{1}}     {{1,1,1},{1}}
                                {{1,1},{2}}     {{1,1},{1,1}}
                               {{1},{1},{2}}   {{1},{1},{1,1}}
                                               {{1},{1,1},{1}}
                                               {{1,1},{1},{1}}
                                              {{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000837, A007716, A055887, A063834, A255397, A269134, A276024, A289508, A316222, A317545, A317546, A319002, A319003.

\\ here B(n) is A000837 as vector.
B(n) = {dirmul(vector(n, k, moebius(k)), vector(n, k, numbpart(k)))}
seq(n) ={my(p=x*Ser(B(n))); Vec(1/prod(g=1, n, 1 - subst(p + O(x*x^(n\g)), x, x^g)))} \\ Andrew Howroyd, Jan 16 2023

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

cp's OEIS Frontend

A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

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A317533 Regular triangle read rows: T(n,k) = number of non-isomorphic multiset partitions of size n and length k.

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A302243 Total weight of the n-th twice-odd-factored multiset partition.

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A317532 Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.

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A339113 Products of primes of squarefree semiprime index (A322551).

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A339112 Products of primes of semiprime index (A106349).

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A317546 Number of multimin partitions of integer partitions of n.

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A317545 Number of multimin factorizations of n.

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