A381717 -id:A381717 - cp's OEIS frontend

A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-sums.

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

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A050326 at a(30) = 4, A050326(30) = 5.
First differs from A339742 at a(42) = 5, A339742(42) = 4.
First differs from A381441 at a(30) = 4, A381441(30) = 5.
First differs from A381633 at a(210) = 10, A381633(210) = 12.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1 with distinct sums of prime indices (A056239).
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.
A multiset partition con be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions with distinct block-sums are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no arrow {1,1,2} -> {4}.

			The prime indices of 120 are {1,1,2,3}, with 3 ways:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
with block-sums: {1,6}, {3,4}, {1,2,4}, so a(120) = 3.
The prime indices of 210 are {1,2,3,4}, with 12 ways:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{1},{2},{3},{4}}
with block-sums: {10}, {1,9}, {2,8}, {3,7}, {4,6}, {3,7}, {4,6}, {1,2,7}, {1,3,6}, {1,4,5}, {2,3,5}, {1,2,3,4}, of which 10 are distinct, so a(210) = 10.

Without distinct block-sums we have A381078 (lower A381454), before sums A050320.
For distinct blocks instead of sums we have A381441, before sums A050326, see A358914.
Before taking sums we had A381633.
Positions of 0 are A381806.
Positions of 1 are A381870, superset of A293511.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A116540, A213242, A213385, A213427, A299202, A300385, A317142, A317143, A318360.

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

A383100 Numbers whose prime indices have no permutation with all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108

Offset: 1

Gus Wiseman, Apr 20 2025

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, sum A056239.

			The prime indices of 18 are {1,2,2}, with permutations (1,2,2), (2,1,2), (2,2,1), with run sums (1,4), (2,1,2), (4,1) respectively, so 18 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
   50: {1,3,3}

For distinct instead of equal run-sums we appear to have A381636, counted by A381717.
For run-lengths instead of sums we have A382879, counted by complement of A383013.
These are the positions of 0 in A382877.
For more than one choice we have A383015.
The complement is A383110, counted by A383098.
Partitions of this type are counted by A383096.
For a unique choice we have A383099, counted by A383095.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351294, A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383094, A383097.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==0&]

A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 126, 168, 207, 266, 343, 428, 534, 675, 832, 1039, 1279, 1575, 1933, 2381, 2881, 3524, 4269, 5179, 6237, 7525, 9033, 10860, 12969, 15512, 18475, 22005, 26105, 30973, 36642, 43325, 51078, 60184, 70769, 83152

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240309 at a(11) = 23, A240309(11) = 25.

First differs from A381990 at a(17) = 126, A381990(17) = 127.

			The partition y = (2,2,1,1,1) can be partitioned into sets in the following ways:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
But none of these is itself a set, so y is counted under a(7).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
For normal multisets see A292432, A292444, A116539.
These partitions are ranked by A293243, complement A382200.
The MM-numbers of these multiset partitions (set of sets) are A302494.
Twice-partitions of this type are counted by A358914.
For distinct sums we have A381990 (ranks A381806), complement A381992 (ranks A382075).
The complement is counted by A382077, unique A382079.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions into distinct sets, complement A050345.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870.

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[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]==0&]],{n,0,9}]

a(19)-a(50) from Bert Dobbelaere, Mar 29 2025

A382076 Number of integer partitions of n whose run-sums are not all equal.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 13, 15, 27, 37, 54, 64, 99, 130, 172, 220, 295, 372, 488, 615, 788, 997, 1253, 1547, 1955, 2431, 3005, 3706, 4563, 5586, 6840, 8332, 10139, 12305, 14879, 17933, 21635, 26010, 31181, 37314, 44581, 53156, 63259, 75163, 89124, 105553, 124752, 147210

Offset: 0

Gus Wiseman, Apr 02 2025

nonn

Also the number of integer partitions of n that cannot be partitioned into distinct constant multisets with a common sum. Multiset partitions of this type are ranked by A005117 /\ A326534 /\ A355743, while twice-partitions are counted by A382524, strict case of A279789.

			The partition (3,2,1,1,1) has runs ((3),(2),(1,1,1)) with sums (3,2,3) so is counted under a(8).
The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)     (43)      (53)
              (41)    (51)     (52)      (62)
              (221)   (321)    (61)      (71)
              (311)   (411)    (322)     (332)
              (2111)  (2211)   (331)     (431)
                      (21111)  (421)     (521)
                               (511)     (611)
                               (2221)    (3221)
                               (3211)    (3311)
                               (4111)    (4211)
                               (22111)   (5111)
                               (31111)   (22211)
                               (211111)  (32111)
                                         (311111)
                                         (2111111)

The complement is counted by A304442, ranks A353833.
For distinct instead of equal block-sums we have A381717.
This is the strict case of A381993, see A381995, zeros A381871.
A050361 counts factorizations into distinct prime powers, see A381715.
A304405 counts partitions with weakly decreasing run-sums, ranks A357875.
A304406 counts partitions with weakly increasing run-sums, ranks A357861.
A304428 counts partitions with strictly decreasing run-sums, ranks A357862.
A304430 counts partitions with strictly increasing run-sums, ranks A357864.
A317141 counts coarsenings of prime indices, refinements A300383.
A326534 ranks multiset partitions with a common sum.
A353837 counts partitions with distinct run-sums.
A354584 lists run-sums of weakly increasing prime indices.
A355743 ranks multiset partitions into constant blocks.
Cf. A000688, A005117, A006171, A047966, A279784, A381453, A381455, A381635, A381636, A381994, A382204.

Table[Length[Select[IntegerPartitions[n],!SameQ@@Total/@Split[#]&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 13, 25, 33, 54, 54, 99, 124, 166, 207, 295, 352, 488, 591, 780, 987, 1253, 1488, 1951, 2419, 2993, 3665, 4563, 5508, 6840, 8270, 10127, 12289, 14869, 17781, 21635, 25992, 31167, 37184, 44581, 53008, 63259, 75076, 89080, 105531, 124752, 146842, 173516, 204141, 239921, 281461, 329929, 385852

Offset: 0

Gus Wiseman, Mar 17 2025

nonn

			The multiset partition {{2},{2},{1,1},{1,1}} has both properties (constant blocks and common sum), so (2,2,1,1,1,1) is not counted under a(8). We can also use {{2,2},{1,1,1,1}}.
The a(3) = 1 through a(8) = 13 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (32111)
                             (211111)  (311111)

Twice-partitions of this type (constant with equal) are counted by A279789.
Multiset partitions of this type are ranked by A326534 /\ A355743.
For distinct instead of equal block-sums we have A381717.
These partitions are ranked by A381871, zeros of A381995.
For strict instead of constant blocks we have A381994, see A381719, A382080.
The strict case is A382076.
Normal multiset partitions of this type are counted by A382204.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000688, A006171, A047966, A265947, A279784, A381453, A381455, A381635, A381636, A381992.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]==0&]],{n,0,30}]

a(31)-a(54) from Robert Price, Mar 31 2025

A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 9, 0, 7, 0, 12, 0, 1, 0, 38, 0, 1, 1, 18, 0, 38, 0, 32, 0, 1, 0, 90, 0, 1, 0, 71, 0, 78, 0, 33, 10, 1, 0, 228, 0, 31, 0, 42, 0, 156, 0, 123, 0, 1, 0, 447, 0, 1, 16, 146, 0, 222, 0, 63, 0, 102, 0, 811, 0, 1, 29, 75, 0, 334, 0

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
  (211)  (3111)  (422)     (511111)  (633)        (71111111)  (844)
                 (41111)             (6222)                   (82222)
                 (221111)            (33222)                  (442222)
                                     (4221111)                (44221111)
                                     (6111111)                (422221111)
                                     (33111111)               (811111111)
                                     (222111111)              (4411111111)
                                                              (42211111111)
                                                              (222211111111)

These partitions are ranked by A383015, positions of terms > 1 in A382877.
For run-lengths instead of sums we have A383090, ranks A383089, unique A383094.
The complement is A383095 + A383096, ranks A383099 \/ A383100.
For any positive number of permutations we have A383098, ranks A383110.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A381717, A382076.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A383110 Numbers whose prime indices have a permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173

Offset: 1

Gus Wiseman, Apr 20 2025

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, sum A056239.

			The prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), so 144 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}

For distinct run-sums we appear to have complement of A381636 (counted by A381717).
These are the positions of positive terms in A382877.
For run-lengths instead of sums we have complement of A382879, counted by A383013.
For more than one choice we have A383015.
Partitions of this type are counted by A383098.
For a unique choice we have A383099, counted by A383095.
The complement is A383100, counted by A383096.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]>0&]

Equals A383015 \/ A383099, counted by A353851 \/ A383095.

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]==0&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

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&]

cp's OEIS Frontend

A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-sums.

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A383100 Numbers whose prime indices have no permutation with all equal run-sums.

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A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.

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A382076 Number of integer partitions of n whose run-sums are not all equal.

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A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

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A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

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A383110 Numbers whose prime indices have a permutation with all equal run-sums.

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A383096 Number of integer partitions of n having no permutation with all equal run-sums.

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

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