A326533 -id:A326533 - cp's OEIS frontend

A007837 Number of partitions of n-set with distinct block sizes.

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847

Offset: 0

Arnold Knopfmacher

nonn

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..700
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Fig. 3, arXiv:math/0606370 [math.CO], 2006.
Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011

nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)

{my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Christian G. Bower

a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A326534 MM-numbers of multiset partitions where every part has the same sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 191

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A298538 in lacking 187.

These are numbers where each prime index has the same sum of prime indices. 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 obtained by taking the multiset of prime indices of each prime index 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}}.

			The sequence of multiset partitions where every part has the same sum, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A035470, A038041, A112798, A302242, A320324, A321455, A326518, A326533, A326535, A326536, A326537.

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

A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 85, 89, 91, 93, 97, 99, 101, 103, 107, 109, 113, 121, 123, 125, 127, 128, 131, 135, 137, 139, 149, 151, 153

Offset: 1

Gus Wiseman, Oct 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The terms together with their corresponding multiset multisystems (A302242):
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
  33: {{1},{3}}
  37: {{1,1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

Amiram Eldar, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A001222, A038041, A112798, A302242, A306017, A317583, A319066, A319169, A320325, A322794, A326533, A326534, A326535, A326536, A326537.

Select[Range[100],SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

is(n) = #Set(apply(p -> bigomega(primepi(p)), factor(n)[,1]~))<=1 \\ Rémy Sigrist, Oct 11 2018

A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 28, 51, 92, 164, 289, 504, 871, 1493, 2539, 4290, 7201, 12017, 19939, 32911, 54044, 88330, 143709, 232817, 375640, 603755, 966816, 1542776, 2453536, 3889338, 6146126, 9683279, 15211881, 23830271, 37230720, 58015116, 90174847, 139820368, 216286593

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

Also the number of integer compositions of n whose leaders of maximal weakly decreasing runs are strictly increasing. For example, the composition (1,2,2,1,3,1,4,1) has maximal weakly decreasing runs ((1),(2,2,1),(3,1),(4,1)), with leaders (1,2,3,4), so is counted under a(15). - Gus Wiseman, Aug 21 2024

			The a(1) = 1 through a(5) = 15 multiset partitions:
  {1}  {2}    {3}        {4}          {5}
       {1,1}  {1,2}      {1,3}        {1,4}
              {1,1,1}    {2,2}        {2,3}
              {1},{1,1}  {1,1,2}      {1,1,3}
                         {1,1,1,1}    {1,2,2}
                         {1},{1,2}    {1,1,1,2}
                         {2},{1,1}    {1},{1,3}
                         {1},{1,1,1}  {1},{2,2}
                                      {2},{1,2}
                                      {3},{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}
From _Gus Wiseman_, Aug 21 2024: (Start)
The a(0) = 1 through a(5) = 15 compositions whose leaders of maximal weakly decreasing runs are strictly increasing:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The version for set partitions is A007837.
For sums instead of sizes we have A271619.
For constant instead of distinct sizes we have A319066.
These multiset partitions are ranked by A326533.
For odd instead of distinct sizes we have A356932.
The version for twice-partitions is A358830.
The case of distinct sums also is A358832.
Ranked by positions of strictly increasing rows in A374740, opposite A374629.
A001970 counts multiset partitions of integer partitions.
A011782 counts compositions.
A063834 counts twice-partitions, strict A296122.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Runs and leaders: A000041, A188900, A374634, A374635, A374637, A374679, A374742 (ranks A374744), A374743 (ranks A374701), A374746, A374747.
Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.
Cf. A003242, A106356, A188920, A189076, A238343, A333213, A336342.

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]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@Length/@#&]],{n,0,10}]
(* second program *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ Andrew Howroyd, Dec 31 2022

G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

A326535 MM-numbers of multiset partitions where each part has a different sum.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A298540 in having 187.

These are numbers where each prime index has a different sum of prime indices. 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 obtained by taking the multiset of prime indices of each prime index 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}}.

			The sequence of multiset partitions where each part has a different sum, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}

Robert Price, Table of n, a(n) for n = 1..9715
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A112798, A275780, A302242, A320324, A321469, A326519, A326533, A326534, A326536, A326537.

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

A188900 Number of compositions of n that avoid the pattern 12-3.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 60, 114, 215, 402, 746, 1375, 2520, 4593, 8329, 15036, 27027, 48389, 86314, 153432, 271853, 480207, 845804, 1485703, 2603018, 4549521, 7933239, 13803293, 23966682, 41530721, 71830198, 124010381, 213725823, 367736268, 631723139, 1083568861

Offset: 0

Nathaniel Johnston, Apr 17 2011

nonn

First differs from the non-dashed version A102726 at a(9) = 215, A102726(9) = 214, due to the composition (1,3,2,3).
The value a(11) = 7464 in Heubach et al. is a typo.
Theorem: A composition avoids 3-12 iff its leaders of maximal weakly decreasing runs are weakly increasing. For example, the composition q = (1,1,2,1,2,2,1,3) has maximal weakly decreasing runs ((1,1),(2,1),(2,2,1),(3)), with leaders (1,2,2,3), which are weakly increasing, so q is counted under a(13); also q avoids 3-12, as required. On the other hand, the composition q = (3,2,1,2,2,1,2) has maximal weakly decreasing runs ((3,2,1),(2,2,1),(2)), with leaders (3,2,2), which are not weakly increasing, so q is not counted under a(13); also q matches 3-12, as required. - Gus Wiseman, Aug 21 2024

			The initial terms are too dense, but see A375406 for the complement. - _Gus Wiseman_, Aug 21 2024

Nathaniel Johnston, Table of n, a(n) for n = 0..200
S. Heubach, T. Mansour and A. O. Munagi, Avoiding Permutation Patterns of Type (2,1) in Compositions, Online Journal of Analytic Combinatorics, 4 (2009).

The non-dashed version A102726, non-ranks A335483.
For 23-1 we have A189076.
The non-ranks are a subset of A335479 and do not include 404, 788, 809, ...
For strictly increasing leaders we have A358836, ranks A326533.
The strict version is A374762.
The complement is counted by A375406.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A188920, A238343, A261982, A333175, A333213, A374630, A374679, A374688, A374740, A374741.

with(PolynomialTools):n:=20:taypoly:=taylor(mul(1/(1 - x^i/mul(1-x^j,j=1..i-1)),i=1..n),x=0,n+1):seq(coeff(taypoly,x,m),m=0..n);

m = 35;
Product[1/(1 - x^i/Product[1 - x^j, {j, 1, i - 1}]), {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Mar 31 2020 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], LessEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

G.f.: Product_{i>=1} (1/(1 - x^i/Product_{j=1..i-1} (1 - x^j))).

a(n) = 2^(n-1) - A375406(n). - Gus Wiseman, Aug 22 2024

A326517 Number of normal multiset partitions of weight n where each part has a different size.

Original entry on oeis.org

1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 12 normal multiset partitions:
  {}  {{1}}  {{1,1}}  {{1,1,1}}
             {{1,2}}  {{1,1,2}}
                      {{1,2,2}}
                      {{1,2,3}}
                      {{1},{1,1}}
                      {{1},{1,2}}
                      {{1},{2,2}}
                      {{1},{2,3}}
                      {{2},{1,1}}
                      {{2},{1,2}}
                      {{2},{1,3}}
                      {{3},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A332253.

Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 23 2023

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_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020

Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020

A382201 MM-numbers of sets of sets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 39, 41, 43, 47, 51, 55, 58, 59, 62, 65, 66, 67, 73, 78, 79, 82, 83, 85, 86, 87, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 129, 130, 134, 137, 139, 141, 145, 146, 149, 155, 157, 158, 163, 165

Offset: 1

Gus Wiseman, Mar 21 2025

nonn

First differs from A302494 in lacking 143, corresponding to the multiset partition {{1,2},{3}}.
Also products of prime numbers of squarefree index such that the factors all have distinct sums of prime indices.
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, sum A056239. 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 terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  22: {{},{3}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}
  34: {{},{4}}
  39: {{1},{1,2}}

Set partitions of this type are counted by A275780.
Twice-partitions of this type are counted by A279785.
For just sets of sets we have A302478.
For distinct blocks instead of block-sums we have A302494.
For equal instead of distinct sums we have A302497.
For just distinct sums we have A326535.
For normal multiset partitions see A326519, A326533, A326537, A381718.
Factorizations of this type are counted by A381633. See also A001055, A045778, A050320, A050326, A321455, A321469, A382080.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A003963, A005117, A007716, A293511, A302242, A319899, A326534, A368100, A368101, A381635, A382215.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@SquareFreeQ/@prix[#]&&UnsameQ@@Total/@prix/@prix[#]&]

Equals A302478 /\ A326535.

A326514 Number of factorizations of n into factors > 1 where each factor has a different number of prime factors counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 5, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 3, 4, 1, 4, 1, 3, 1, 4, 1, 6, 1, 1, 3, 3, 1, 4, 1, 5, 2, 1, 1, 4, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 11 2019

nonn

			The a(96) = 8 factorizations: (2*4*12), (2*6*8), (2*48), (3*4*8), (3*32), (4*24), (6*16), (96).

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A001055, A007837, A038041, A321455, A321469, A322794, A326026, A326515, A326516, A326517, A326533.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@PrimeOmega/@#&]],{n,100}]

A326537 MM-numbers of multiset partitions where each part has a different average.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

These are numbers where each prime index has a different average of prime indices. 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 obtained by taking the multiset of prime indices of each prime index 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}}.

			The sequence of multiset partitions where each part has a different average, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A051293, A112798, A302242, A320324, A326513, A326516, A326521, A326533, A326534, A326535, A326536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Mean/@primeMS/@primeMS[#]&]

cp's OEIS Frontend

A007837 Number of partitions of n-set with distinct block sizes.

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A326534 MM-numbers of multiset partitions where every part has the same sum.

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A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

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A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

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A326535 MM-numbers of multiset partitions where each part has a different sum.

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A188900 Number of compositions of n that avoid the pattern 12-3.

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A326517 Number of normal multiset partitions of weight n where each part has a different size.

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