A326514 -id:A326514 - 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

A326622 Number of factorizations of n into factors > 1 with integer average.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 3, 2, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 5, 2, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 3, 3, 1, 1, 6, 2, 2, 2, 2, 1, 2, 2, 4, 2, 1, 1, 6, 1, 1, 3, 7, 2, 1, 1, 3, 2, 1, 1, 6, 1, 1, 3, 2, 2, 2, 1, 7, 5, 1, 1, 4, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 8, 1, 1, 3, 3, 1, 1, 1, 4, 5, 1, 1, 6

Offset: 1

Gus Wiseman, Jul 14 2019

nonn

			The a(80) = 7 factorizations:
  (2*2*2*10)
  (2*2*20)
  (2*5*8)
  (2*40)
  (4*20)
  (8*10)
  (80)

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

Partitions with integer average are A067538.
Strict partitions with integer average are A102627.
Heinz numbers of partitions with integer average are A316413.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A051293, A078174, A078175, A326514, A326515, A326567/A326568, A326621, A326623, A326667 (= a(2^n)), A327906 (positions of 1's), A327907 (of terms > 1).

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],IntegerQ[Mean[#]]&]],{n,2,100}]

A326622(n, m=n, facsum=0, facnum=0) = if(1==n,facnum > 0 && 1==denominator(facsum/facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326622(n/d, d, facsum+d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024

Data section extended up to a(108), with missing term a(1)=0 also added (thus correcting the offset) - Antti Karttunen, Nov 10 2024

A321469 Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 2, 4, 2, 5, 1, 3, 2, 4, 1, 8, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 7, 2, 2, 2, 5, 1, 7, 2, 3, 2, 2, 2, 8, 1, 3, 3, 5, 1, 5, 1, 5, 5

Offset: 1

Gus Wiseman, Nov 11 2018

nonn

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 sum of prime indices of n is A056239(n).

			The a(72) = 8 multiset partitions with distinct block-sums:
    {{1,1,1,2,2}}
   {{1},{1,1,2,2}}
   {{2},{1,1,1,2}}
   {{1,1},{1,2,2}}
   {{1,2},{1,1,2}}
   {{2,2},{1,1,1}}
  {{1},{2},{1,1,2}}
  {{1},{1,1},{2,2}}
Missing from this list are:
    {{1},{1},{1,2,2}}
    {{1},{1,2},{1,2}}
    {{2},{2},{1,1,1}}
    {{2},{1,1},{1,2}}
   {{1},{1},{1},{2,2}}
   {{1},{1},{2},{1,2}}
   {{1},{2},{2},{1,1}}
  {{1},{1},{1},{2},{2}}

Cf. A001055, A056239, A275780, A317144, A321455, A322794, A326514, A326515, A326516, A326519, A326535, A371734.

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

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A321469(n, m=n, facs=List([])) = if(1==n, all_have_different_sum_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A321469(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A326028 Number of factorizations of n into factors > 1 with integer geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jul 15 2019

nonn

First differs from A294336 and A316782 at a(36) = 5.

			The a(4) = 2 through a(36) = 5 factorizations (showing only the cases where n is a perfect power).
  (4)    (8)      (9)    (16)       (25)   (27)     (32)         (36)
  (2*2)  (2*2*2)  (3*3)  (2*8)      (5*5)  (3*3*3)  (2*2*2*2*2)  (4*9)
                         (4*4)                                   (6*6)
                         (2*2*2*2)                               (2*18)
                                                                 (3*12)

Antti Karttunen, Table of n, a(n) for n = 1..100000
Wikipedia, Geometric mean
Index entries for sequences computed from exponents in factorization of n

Positions of terms > 1 are the perfect powers A001597.
Partitions with integer geometric mean are A067539.
Subsets with integer geometric mean are A326027.
Factorizations with integer average and geometric mean are A326647.
Cf. A001055, A082553, A322794, A326514, A326515, A326516, A326622, A326623, A326624, A326625.

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],IntegerQ[GeometricMean[#]]&]],{n,2,100}]

A326028(n, m=n, facmul=1, facnum=0) = if(1==n,facnum>0 && ispower(facmul,facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326028(n/d, d, facmul*d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024

a(2^n) = A067538(n).

a(89) onwards from Antti Karttunen, Nov 10 2024

A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 7, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

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 a(900) = 9 factorizations:
  (3*3*10*10),
  (3*3*100), (3*10*30), (9*10*10),
  (3*300), (9*100), (10*90), (30*30),
  (900).

Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326512, A326514, A326516, A326520, A326536.

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]]}]];
Table[Length[Select[facs[n],SameQ@@Mean/@primeMS/@#&]],{n,100}]

avgpis(n) = { my(f=factor(n)); f[,1] = apply(primepi,f[,1]); (1/bigomega(n))*sum(i=1,#f~,f[i,2]*f[i,1]); };
has_same_average_of_pis(facs) = if(!#facs, 1, my(avg=0); for(i=1,#facs,if(!avg, avg=avgpis(facs[i]), if(avg!=avgpis(facs[i]), return(0)))); (1));
A326515(n, m=n, facs=List([])) = if(1==n, has_same_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326515(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

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

A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 26 2018

nonn

Also the number of uniform multiset partitions of the multiset of prime indices of n, where a multiset partition is uniform if all parts have the same size.

			The a(1260) = 13 factorizations:
  (1260)  (18*70)   (4*9*35)   (2*2*3*3*5*7)
          (20*63)   (6*6*35)
          (28*45)   (4*15*21)
          (30*42)   (6*10*21)
          (12*105)  (6*14*15)
                    (9*10*14)
The a(1260) = 13 multiset partitions:
  {{1},{1},{2},{2},{3},{4}}
     {{1,1},{2,2},{3,4}}
     {{1,1},{2,3},{2,4}}
     {{1,2},{1,2},{3,4}}
     {{1,2},{1,3},{2,4}}
     {{1,2},{1,4},{2,3}}
     {{2,2},{1,3},{1,4}}
      {{1,1,2},{2,3,4}}
      {{1,2,2},{1,3,4}}
      {{1,1,3},{2,2,4}}
      {{1,1,4},{2,2,3}}
      {{1,2,3},{1,2,4}}
       {{1,1,2,2,3,4}}

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

Cf. A001055, A001222, A038041, A112798, A306017, A306021, A319169, A320324, A317583, A321455, A321469, A326514, A326515, A326516.

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],SameQ@@PrimeOmega/@#&]],{n,100}]

A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 8, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 8, 2, 2, 2, 4, 1, 7, 2, 3, 2, 2, 2, 6, 1, 3, 3, 5, 1, 5, 1, 4, 4

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

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 a(60) = 8 factorizations: (2*5*6), (3*4*5), (2*30), (3*20), (4*15), (5*12), (6*10), (60).

Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326513, A326514, A326515, A326521, A326537.

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]]}]];
Table[Length[Select[facs[n],UnsameQ@@Mean/@primeMS/@#&]],{n,100}]

avgpis(n) = { my(f=factor(n)); f[,1] = apply(primepi,f[,1]); (1/bigomega(n))*sum(i=1,#f~,f[i,2]*f[i,1]); };
all_have_different_average_of_pis(facs) = if(!#facs, 1, (#Set(apply(avgpis,facs)) == #facs));
A326516(n, m=n, facs=List([])) = if(1==n, all_have_different_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326516(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A326647 Number of factorizations of n into factors > 1 with integer average and integer geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 16 2019

nonn

			The a(216) = 5 factorizations:
  (2*4*27)
  (3*3*24)
  (3*6*12)
  (6*6*6)
  (216)
The a(729) = 8 factorizations:
  (3*3*3*3*3*3)
  (3*3*81)
  (3*9*27)
  (3*243)
  (9*9*9)
  (9*81)
  (27*27)
  (729)

Wikipedia, Geometric mean

Positions of terms > 1 are the perfect powers A001597.
Factorizations with integer average are A326622.
Factorizations with integer geometric mean are A326028.
Partitions with integer average and geometric mean are A326641.
Cf. A001055, A067538, A067539, A322794, A326514, A326515, A326516, A326623, A326643, A326645.

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],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,2,100}]

A326533 MM-numbers of multiset partitions where each part has a different length.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 53, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119, 122

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

These are numbers where each prime index has a different Omega (number of prime factors counted with multiplicity). 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}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  31: {{5}}
  34: {{},{4}}
  35: {{2},{1,1}}

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

A subsequence of A005117.

Cf. A007837, A038041, A112798, A302242, A320324, A326026, A326514, A326517, A326534, A326535, A326536, A326537.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A326622 Number of factorizations of n into factors > 1 with integer average.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A321469 Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A326028 Number of factorizations of n into factors > 1 with integer geometric mean.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.

Views

Author

Keywords