A293511 -id:A293511 - cp's OEIS frontend

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 9, 13, 14, 21, 20, 32, 31, 42, 47, 63, 62, 90, 94, 117, 138, 170, 186, 235, 260, 315, 363, 429, 493, 588, 674, 795, 901, 1060, 1209, 1431, 1608, 1896, 2152, 2515, 2854, 3310, 3734, 4368, 4905, 5686

Offset: 0

Gus Wiseman, Mar 20 2025

			The unique multiset partition for (3222111) is {{1},{2},{1,2},{1,2,3}}.
The a(1) = 1 through a(12) = 13 partitions:
  1  2  3  4    5    6     7    8      9      A      B      C
           211  221  411   322  332    441    433    443    552
                311  2211  331  422    522    442    533    633
                           511  611    711    622    551    822
                                3311   42111  811    722    A11
                                32111         3322   911    4422
                                              4411   42221  5511
                                              32221  53111  33321
                                              43111  62111  52221
                                              52111         54111
                                                            63111
                                                            72111
                                                            3222111

Normal multiset partitions of this type are counted by A116539, see A381718.
These partitions are ranked by A293511.
MM-numbers of these multiset partitions (sets of sets) are A302494, see A302478, A382201.
Twice-partitions of this type (sets of sets) are counted by A358914, see A279785.
For at least one choice we have A382077 (ranks A382200), see A381992 (ranks A382075).
For no choices we have A382078 (ranks A293243), see A381990 (ranks A381806).
For distinct block-sums instead of blocks we have A382460, ranked by A381870.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets, see A381633.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A213427, A299202, A317142, A381078, A381441, A381454, A381636.

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[ssfacs[Times@@Prime/@#]]==1&]],{n,0,15}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A339841 Numbers that can be factored into distinct primes or semiprimes in exactly one way.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 48, 49, 53, 59, 61, 67, 71, 73, 79, 80, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 131, 137, 139, 144, 149, 151, 157, 162, 163, 167, 169, 173, 176, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Dec 25 2020

nonn

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

			The sequence of terms together with their one factorization begins:
     1 =        29 = 29        80 = 2*4*10
     2 = 2      31 = 31        83 = 83
     3 = 3      37 = 37        89 = 89
     4 = 4      41 = 41        97 = 97
     5 = 5      43 = 43       101 = 101
     7 = 7      47 = 47       103 = 103
     8 = 2*4    48 = 2*4*6    107 = 107
     9 = 9      49 = 49       109 = 109
    11 = 11     53 = 53       112 = 2*4*14
    13 = 13     59 = 59       113 = 113
    17 = 17     61 = 61       121 = 121
    19 = 19     67 = 67       125 = 5*25
    23 = 23     71 = 71       127 = 127
    25 = 25     73 = 73       131 = 131
    27 = 3*9    79 = 79       137 = 137
For example, we have 360 = 2*3*6*10, so 360 is in the sequence. But 360 is absent from A293511, because we also have 360 = 2*6*30.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross-references.
These are the positions of ones in A339839.
The version for no factorizations is A339840.
The version for at least one factorization is A339889.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A037143 lists primes and semiprimes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A338915 counts partitions that cannot be partitioned into distinct pairs.
Cf. A002494, A013929, A028260, A320893, A320922, A339618, A339740, A339846.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Length[Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]]==1&]

A339839 Number of factorizations of n into distinct primes or semiprimes.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

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

			The a(n) factorizations for n = 6, 16, 30, 60, 180, 210, 240, 420:
  6    5*6    4*15    4*5*9    6*35     4*6*10    2*6*35
  2*3  2*15   6*10    2*6*15   10*21    2*4*5*6   3*4*35
       3*10   2*5*6   2*9*10   14*15    2*3*4*10  4*5*21
       2*3*5  3*4*5   3*4*15   5*6*7              4*7*15
              2*3*10  3*6*10   2*3*35             5*6*14
                      2*3*5*6  2*5*21             6*7*10
                               2*7*15             2*10*21
                               3*5*14             2*14*15
                               3*7*10             2*5*6*7
                               2*3*5*7            3*10*14
                                                  3*4*5*7
                                                  2*3*5*14
                                                  2*3*7*10

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

A008966 allows only primes.
A320732 is the non-strict version.
A339742 does not allow squares of primes.
A339840 lists the positions of zeros.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A013929 cannot be factored into distinct primes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A322353 into distinct semiprimes.
- A339839 [this sequence] into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A000569 counts graphical partitions (A320922).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000070, A001222, A028260, A320893, A338898, A339617, A339661, A339740, A339741, A339846.

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@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]],{n,100}]

A339839(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA339839(n/d, d))); (s)); \\ Antti Karttunen, Feb 10 2023

a(n) = Sum_{d|n squarefree} A322353(n/d).

Data section extended up to a(105) by Antti Karttunen, Feb 10 2023

A300442 Number of binary strict trees of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 23, 46, 108, 231, 561, 1285, 3139, 7348, 18265, 43907, 109887, 267582, 675866, 1669909, 4238462, 10555192, 26955062, 67706032, 173591181, 438555624, 1129088048, 2869732770, 7410059898, 18911818801, 48986728672, 125562853003, 326011708368

Offset: 0

Gus Wiseman, Mar 05 2018

nonn

A binary strict tree of weight n > 0 is either a single node of weight n, or an ordered pair of binary strict trees with strictly decreasing weights summing to n.

			The a(5) = 6 binary strict trees: 5, (41), (32), ((31)1), ((21)2), (((21)1)1).
The a(6) = 10 binary strict trees:
  6,
  (51), (42),
  ((41)1), ((32)1), ((31)2),
  (((31)1)1), (((21)2)1), (((21)1)2),
  ((((21)1)1)1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000992, A001190, A063834, A196545, A273873, A289501, A292432, A293511, A300352, A300440, A300443.

a:= proc(n) option remember;
      1+add(a(j)*a(n-j), j=1..(n-1)/2)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 06 2018

k[n_]:=k[n]=1+Sum[Times@@k/@y,{y,Select[IntegerPartitions[n],Length[#]===2&&UnsameQ@@#&]}];
Array[k,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n - j], {j, 1, (n - 1)/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, (n-1)\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018

a(n) = 1 + Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Apr 22 2025

nonn

Differs from A059404, A323055, A376250 in lacking 150.
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.
Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).

			The prime indices of 150 are {1,2,3,3}, and {{3},{3},{1,2}} is a partition into sets with a common sum, so 150 is not in the sequence.

Twice-partitions of this type (sets with a common sum) are counted by A279788.
These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.
For distinct block-sums we have A381806, counted by A381990 (complement A381992).
For constant blocks we have A381871 (zeros of A381995), counted by A381993.
Partitions of this type are counted by A381994.
These are the zeros of A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
The complement counted by A383308.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A381633 counts set systems with distinct sums, see A381634, A293243.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.

prix[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]]]];
Select[Range[100],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Mar 22 2025

nonn

Also numbers with a unique factorization into prime powers with 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 prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

For distinct blocks instead of block-sums we have A004709, counted by A000726.
Twice-partitions of this type are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
These are the positions of 1 in A381635.
For no choices we have A381636 (zeros of A381635), counted by A381717.
For strict instead of constant blocks we have A381870, counted by A382079.
Partitions of this type (unique into constant with distinct) are counted by A382301.
Normal multiset partitions of this type are counted by A382203.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
Cf. A006171, A279784, A295935.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]

A339840 Numbers that cannot be factored into distinct primes or semiprimes.

Original entry on oeis.org

16, 32, 64, 81, 96, 128, 160, 192, 224, 243, 256, 288, 320, 352, 384, 416, 448, 486, 512, 544, 576, 608, 625, 640, 704, 729, 736, 768, 800, 832, 864, 896, 928, 960, 972, 992, 1024, 1088, 1152, 1184, 1215, 1216, 1280, 1312, 1344, 1376, 1408, 1458, 1472, 1504

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

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

			The sequence of terms together with their prime indices begins:
    16: {1,1,1,1}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    96: {1,1,1,1,1,2}
   128: {1,1,1,1,1,1,1}
   160: {1,1,1,1,1,3}
   192: {1,1,1,1,1,1,2}
   224: {1,1,1,1,1,4}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   288: {1,1,1,1,1,2,2}
   320: {1,1,1,1,1,1,3}
   352: {1,1,1,1,1,5}
   384: {1,1,1,1,1,1,1,2}
   416: {1,1,1,1,1,6}
   448: {1,1,1,1,1,1,4}
   486: {1,2,2,2,2,2}
For example, a complete list of all factorizations of 192 into primes or semiprimes is:
  (2*2*2*2*2*2*3)
  (2*2*2*2*2*6)
  (2*2*2*2*3*4)
  (2*2*2*4*6)
  (2*2*3*4*4)
  (2*4*4*6)
  (3*4*4*4)
Since none of these is strict, 192 is in the sequence.

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

Allowing only primes gives A013929.
Removing all squares of primes gives A339740.
These are the positions of zeros in A339839.
The complement is A339889.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A338915 cannot be partitioned into distinct pairs (A320892).
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339742 into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n, ranked by A339618.
- A339655 counts non-loop-graphical partitions of 2n (A339657).
Cf. A000070, A028260, A320893, A320922, A339741, A339846.

filter:= proc(n)
  g(map(t -> t[2], ifactors(n)[2]))
end proc;
g:= proc(L) option remember; local x,i,j,t,s,Cons,R;
  if nops(L) = 1 then return L[1] > 3
  elif nops(L) = 2 then return max(L) > 4
  fi;
  Cons:= {seq(x[i] + x[i,i] + add(x[j,i], j=1..i-1)
     + add(x[i,j],j=i+1..nops(L)) = L[i], i=1..nops(L))};
  R:= traperror(Optimization:-LPSolve(0,Cons, assume=binary));
  type(R,string)
end proc:
select(filter, [$2..2000]); # Robert Israel, Dec 28 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]=={}&]

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}]

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}]

A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 0, 7, 3, 11, 18, 9

Offset: 0

Gus Wiseman, Mar 30 2025

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 {1,2,2,2,2,3,3,4} has three multiset partitions into a set of sets:
  {{2},{1,2},{2,3},{2,3,4}}
  {{2},{2,3},{2,4},{1,2,3}}
  {{2},{3},{1,2},{2,3},{2,4}}
so is not counted under a(8).
The a(1) = 1 through a(7) = 7 normal multisets:
  {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
          {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
                              {1,2,3,3,3}     {1,1,1,2,3,3,3}
                                              {1,2,2,2,2,3,4}
                                              {1,2,2,2,3,3,3}
                                              {1,2,3,3,3,3,4}
                                              {1,2,3,4,4,4,4}

For constant instead of strict blocks we have A000045.
Factorizations of this type are counted by A050326, with distinct sums A381633.
For the strong case see A292444, A382430, complement A381996, A382523.
MM-numbers of sets of sets are A302494, see A302478, A382201.
Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.
For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).
With distinct sums we have A382459.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, A382428.

allnorm[n_]:=If[n<=0,{{}},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[#], UnsameQ@@#&&And@@UnsameQ@@@#&]]==1&]], {n,0,5}]

cp's OEIS Frontend

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A339841 Numbers that can be factored into distinct primes or semiprimes in exactly one way.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A339839 Number of factorizations of n into distinct primes or semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A300442 Number of binary strict trees of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A339840 Numbers that cannot be factored into distinct primes or semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Examples