A050326 -id:A050326 - cp's OEIS frontend

A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

1, 1, 1, 6, 8, 35, 292, 673, 2818, 16956, 219772, 636748, 3768505, 20309534, 183403268, 3227600747, 12272598308, 81353466578, 561187259734, 4416808925866, 50303004612136, 1238783066956740, 5566249468690291, 44970939483601100, 330144217684933896, 3131452652308459402

Offset: 0

Gus Wiseman, Mar 29 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 8 multiset partitions:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
                  {{1},{1,2}}  {{1},{1,2,3}}
                  {{1},{2,3}}  {{1},{2,3,4}}
                  {{2},{1,2}}  {{2},{1,2,3}}
                  {{2},{1,3}}  {{2},{1,3,4}}
                  {{3},{1,2}}  {{3},{1,2,3}}
                               {{3},{1,2,4}}
                               {{4},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

For distinct sums instead of sizes we have A116539, see A050326.
Without distinct lengths we have A116540 (normal set multipartitions).
Without strict blocks we have A326517, for sum instead of size A326519.
For equal instead of distinct sizes we have A331638.
Twice-partitions of this type are counted by A358830.
For distinct sums instead of sizes we have A381718.
For equal instead of distinct sizes we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A279785, A317532, A326518, A381633, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Length/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k, 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, Mar 31 2025

a(10) onwards from Andrew Howroyd, Mar 31 2025

A179695 Numbers of the form p^3q^2r^2 where p, q, and r are distinct primes.

Original entry on oeis.org

1800, 2700, 3528, 4500, 5292, 8712, 9800, 12168, 12348, 13068, 18252, 20808, 24200, 24500, 25992, 31212, 33075, 33800, 34300, 38088, 38988, 47432, 47916, 55125, 57132, 57800, 60500, 60552, 66248, 69192, 72200, 77175, 79092, 81675, 84500

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

Subsequence of A225228. - Reinhard Zumkeller, May 03 2013

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of prime signatures, 2010.
Index to sequences related to prime signature.

Cf. A050326, A225228.

Cf. A085548, A085541, A085964, A085965, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,3}; Select[Range[10^5], f]
f[n_]:={Times@@(n^{2,2,3}),Times@@(n^{2,3,2}),Times@@(n^{3,2,2})}; Module[ {nn=20},Select[Flatten[f/@Subsets[Prime[Range[nn]],{3}]],#<= 72*Prime[ nn]^2&]]//Union (* Harvey P. Dale, Jul 05 2019 *)

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/3), t1=p^3;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=q+1, sqrt(lim\t2), if(p==r,next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179695(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=isqrt(x//r**3))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,7)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

A050326(a(n)) = 5. - Reinhard Zumkeller, May 03 2013

Sum_{n>=1} 1/a(n) = P(2)^2*P(3)/2 - P(3)*P(4)/2 - P(2)*P(5) + P(7) = 0.0032578591481263202818..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 10, 13, 15, 22, 20, 32, 32, 43, 49, 65, 64, 92, 96, 121, 140, 173, 192

Offset: 0

Gus Wiseman, Mar 29 2025

			The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)
The a(1) = 1 through a(10) = 10 partitions (A=10):
  1  2  3  4    5    6     7    8      9      A
           211  221  411   322  332    441    433
                311  2211  331  422    522    442
                           511  611    711    622
                                3311   42111  811
                                32111         3322
                                              4411
                                              32221
                                              43111
                                              52111

Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633.
Normal multiset partitions of this type are counted by A381718.
These partitions are ranked by A381870.
For no choices we have A381990, ranks A381806, see A382078, ranks A293243.
For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.
For distinct blocks instead of block-sums we have A382079, ranks A293511.
MM-numbers of these multiset partitions are A382201, see A302478.
For constant instead of strict blocks we have A382301, ranks A381991.
Set systems: A050320, A050326, A050342, A116539, A296120, A318361.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&) /@ Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,15}]

A190108 Numbers with prime factorization pqr^3*s^3 (where p, q, r, s are distinct primes).

Original entry on oeis.org

7560, 11880, 14040, 16632, 18360, 19656, 20520, 21000, 24840, 25704, 28728, 30888, 31320, 33000, 33480, 34776, 39000, 39960, 40392, 41160, 43848, 44280, 45144, 46440, 46872, 47250, 47736, 50760, 51000, 53352, 54648, 55944, 57000, 57240, 61992, 63720, 64584

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

A050326(a(n)) = 11. - Reinhard Zumkeller, May 03 2013

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*(3^3)*5*7 = 7560;
a(2) = (2^3)*(3^3)*5*11 = 11880;
a(3) = (2^3)*(3^3)*5*13 = 14040;
a(4) = (2^3)*(3^3)*7*11 = 16632.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A179704, A190106, A190107.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3,3};Select[Range[150000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\120, 3), t1=p^3; forprime(q=2,sqrtnint(lim\(6*t1), 3), if(q==p, next); t2=q^3*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Name edited by Petros Hadjicostas, Oct 26 2019

A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 180, 182, 186, 190, 195, 222, 230, 231, 238, 246, 252, 255, 258, 266, 273, 282, 285, 286, 290, 300, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 396, 399, 402, 406, 410, 418, 426, 429

Offset: 1

Reinhard Zumkeller, May 03 2013

nonn

Union of A007304, A179643 and A179695; subsequence of A033992;
A001221(a(n)) = 3 and A051903(a(n)) <= A051904(a(n)) + 1 and A001222(a(n)) = 3 or 5 or 7;
A050326(a(n)) = 5.

			A007304(1) = 2*3*5 = 30, A206778(30,1..8)=[1,2,3,5,6,10,15,30]:
A050326(30) = #{30, 15*2, 10*3, 6*5, 5*3*2} = 5;
A179643(1) = 2^2*3^2*5 = 180, A206778(180,1..8)=[1,2,3,5,6,10,15,30]:
A050326(180) = #{30*6, 30*3*2, 15*6*2, 10*6*3, 6*5*3*2} = 5;
A179695(1) = 2^3*3^2*5^2 = 1800, A206778(1800,1..8)=[1,2,3,5,6,10,15,30]:
A050326(1800) = #{30*10*6, 30*6*5*2, 30*10*3*2, 15*10*6*2, 10*6*5*3*2} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A124010.

a225228 n = a225228_list !! (n-1)
a225228_list = filter f [1..] where
   f x = length es == 3 && sum es `elem` [3,5,7] &&
                           maximum es - minimum es <= 1
         where es = a124010_row x

is(n)=my(f=vecsort(factor(n)[,2]~)); f==[1,1,1] || f==[1,2,2] || f==[2,2,3] \\ Charles R Greathouse IV, Jul 28 2016

a(n) ~ 2n log n / (log log n)^2. - Charles R Greathouse IV, Jul 28 2016

A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 17, 27, 43, 46, 82, 103, 133, 181, 258, 295

Offset: 0

Gus Wiseman, Mar 17 2025

nonn

			For y = (3,3,1,1) we have {{1,3},{1,3}}, so y is not counted under a(8).
For y = (3,2,2,1), although we have {{1,3},{2,2}}, the block {2,2} is not a set, so y is counted under a(8).
The a(4) = 1 through a(8) = 12 partitions:
  (2,1,1)  (2,2,1)    (4,1,1)      (3,2,2)        (3,3,2)
           (3,1,1)    (3,1,1,1)    (3,3,1)        (4,2,2)
           (2,1,1,1)  (2,1,1,1,1)  (5,1,1)        (6,1,1)
                                   (2,2,2,1)      (3,2,2,1)
                                   (3,2,1,1)      (4,2,1,1)
                                   (4,1,1,1)      (5,1,1,1)
                                   (2,2,1,1,1)    (2,2,2,1,1)
                                   (3,1,1,1,1)    (3,2,1,1,1)
                                   (2,1,1,1,1,1)  (4,1,1,1,1)
                                                  (2,2,1,1,1,1)
                                                  (3,1,1,1,1,1)
                                                  (2,1,1,1,1,1,1)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279788.
Interchanging "constant" with "strict" gives A381717, see A381635, A381636, A381991.
Normal multiset partitions of this type are counted by A381718, see A279785.
These partitions are ranked by A381719, zeros of A382080.
For distinct instead of equal block-sums we have A381990, ranked by A381806.
For constant instead of strict blocks we have A381993.
A000041 counts integer partitions, strict A000009.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A381633 counts set systems with distinct sums, see A381634, A293243.
Cf. A002846, A047966, A279786, A279789, A293511, A299202, A317142, A358914, A381992.

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[IntegerPartitions[n], Length[Select[mps[#], And@@UnsameQ@@@#&&SameQ@@Total/@#&]]==0&]],{n,0,10}]

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A292432 at a(9) = 48, A292432(9) = 46.

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {1,1,1,2,2,2}
                                          {1,1,2,2,2,2}
                                          {1,2,2,2,2,2}
                                          {1,2,2,2,2,3}
                                          {1,2,3,3,3,3}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Without distinct sums we have A292432, complement A382214.
The strongly normal version without distinct sums is A292444, complement A381996.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
The complement is counted by A382216.
The strongly normal version is A382430, complement A382460.
The case of a unique choice is counted by A382459, without distinct sums A382458.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,5}]

A382430 Number of non-isomorphic finite multisets of size n that cannot be partitioned into sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 6, 9, 12, 17, 22, 32

Offset: 0

Gus Wiseman, Apr 01 2025

We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.

			The a(2) = 1 through a(7) = 6 multisets:
  {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}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}  {1,1,1,1,1,1,2}
                             {1,1,1,2,2}  {1,1,1,1,2,2}  {1,1,1,1,1,2,2}
                                          {1,1,1,1,2,3}  {1,1,1,1,1,2,3}
                                          {1,1,1,2,2,2}  {1,1,1,1,2,2,2}
                                                         {1,1,1,1,2,2,3}

Twice-partitions of this type are counted by A279785, strict A358914.
The strict version is A292444.
Factorizations of this type are counted by A381633, strict A050326.
Normal multiset partitions of this type are counted by A381718, strict A116539.
For integer partitions we have A381990, ranks A381806, complement A381992, ranks A382075.
The strict version for integer partitions is A382078, ranks A293243, complement A382077, ranks A382200.
The normal version is A382202, complement A382216, strict A292432, complement A382214.
The complement is counted by A382523, strict A381996.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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[strnorm[n],Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]],{n,0,5}]

A190107 Numbers with prime factorization pqr^2s^4.

Original entry on oeis.org

5040, 7920, 8400, 9360, 11088, 11340, 11760, 12240, 13104, 13200, 13680, 15600, 16560, 17136, 17820, 19152, 20400, 20592, 20880, 21060, 22320, 22800, 23184, 24948, 25872, 26640, 26928, 27540, 27600, 28350, 29040, 29232, 29484, 29520

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

A050326(a(n)) = 3. - Reinhard Zumkeller, May 03 2013

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A111467, A179704, A190106.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,4};Select[Range[60000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\60, 4), t1=p^4; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A318371 Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(24) = 6 strict set multipartitions of {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1},{2},{1,3,4}}
  {{1},{1,2},{3,4}}
  {{2},{1,3},{1,4}}
  {{1},{2},{3},{1,4}}

Cf. A007716, A045778, A049311, A050326, A116539, A116540, A283877, A292432 A292444.

Cf. A318283, A318285, A318287, A318360, A318361, A318362, A318369, A318370.

a(n) = A318370(A181821(n)).

cp's OEIS Frontend

A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

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A179695 Numbers of the form p^3*q^2*r^2 where p, q, and r are distinct primes.

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A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

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A190108 Numbers with prime factorization p*q*r^3*s^3 (where p, q, r, s are distinct primes).

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A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

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A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

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A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

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A382430 Number of non-isomorphic finite multisets of size n that cannot be partitioned into sets with distinct sums.

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Examples

A179695 Numbers of the form p^3q^2r^2 where p, q, and r are distinct primes.

A190108 Numbers with prime factorization pqr^3*s^3 (where p, q, r, s are distinct primes).