A320324 -id:A320324 - cp's OEIS frontend

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The case of compositions is A358335, strictly decreasing A358901.
The complement is counted by A358910.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A321698 MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 41, 43, 47, 49, 51, 53, 55, 59, 64, 67, 73, 79, 81, 83, 85, 93, 97, 101, 103, 109, 113, 121, 123, 125, 127, 128, 131, 137, 139, 149, 151, 155, 157, 161, 163, 165, 167, 169, 177, 179

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform and regular, so its MM-number 15463 belongs to the sequence.

			The sequence of all uniform regular multiset multisystems, together with their MM-numbers, begins:
   1: {}                   33: {{1},{3}}            109: {{10}}
   2: {{}}                 41: {{6}}                113: {{1,2,3}}
   3: {{1}}                43: {{1,4}}              121: {{3},{3}}
   4: {{},{}}              47: {{2,3}}              123: {{1},{6}}
   5: {{2}}                49: {{1,1},{1,1}}        125: {{2},{2},{2}}
   7: {{1,1}}              51: {{1},{4}}            127: {{11}}
   8: {{},{},{}}           53: {{1,1,1,1}}          128: {{},{},{},{},{},{}}
   9: {{1},{1}}            55: {{2},{3}}            131: {{1,1,1,1,1}}
  11: {{3}}                59: {{7}}                137: {{2,5}}
  13: {{1,2}}              64: {{},{},{},{},{},{}}  139: {{1,7}}
  15: {{1},{2}}            67: {{8}}                149: {{3,4}}
  16: {{},{},{},{}}        73: {{2,4}}              151: {{1,1,2,2}}
  17: {{4}}                79: {{1,5}}              155: {{2},{5}}
  19: {{1,1,1}}            81: {{1},{1},{1},{1}}    157: {{12}}
  23: {{2,2}}              83: {{9}}                161: {{1,1},{2,2}}
  25: {{2},{2}}            85: {{2},{4}}            163: {{1,8}}
  27: {{1},{1},{1}}        93: {{1},{5}}            165: {{1},{2},{3}}
  29: {{1,3}}              97: {{3,3}}              167: {{2,6}}
  31: {{5}}               101: {{1,6}}              169: {{1,2},{1,2}}
  32: {{},{},{},{},{}}    103: {{2,2,2}}            177: {{1},{7}}

Cf. A005176, A007016, A112798, A271103, A283877, A299353, A302242, A306017, A319056, A319189, A320324, A321699, A321717, A322554, A322703, A322833.

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

A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

			The a(1) = 1 through a(6) = 17 multiset partitions:
  {1}  {2}     {3}        {4}           {5}              {6}
       {11}    {12}       {13}          {14}             {15}
       {1}{1}  {111}      {22}          {23}             {24}
               {1}{1}{1}  {112}         {113}            {33}
                          {1111}        {122}            {114}
                          {2}{2}        {1112}           {123}
                          {11}{11}      {11111}          {222}
                          {1}{1}{1}{1}  {1}{1}{1}{1}{1}  {1113}
                                                         {1122}
                                                         {3}{3}
                                                         {11112}
                                                         {111111}
                                                         {12}{12}
                                                         {2}{2}{2}
                                                         {111}{111}
                                                         {11}{11}{11}
                                                         {1}{1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For just constant sums we have A305551, ranked by A326534.
For just constant lengths we have A319066, ranked by A320324.
The version for set partitions is A327899.
For distinct instead of constant lengths and sums we have A358832.
The version for twice-partitions is A358833.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions, strict A296122.
Cf. A000219, A007425, A141199, A327908, A356932, A358831.

Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]

P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(41) and beyond from Andrew Howroyd, Dec 31 2022

A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 41, 56, 84, 113, 164, 218, 306, 401, 547, 711, 949, 1218, 1599, 2034, 2625, 3310, 4224, 5283, 6664, 8271, 10336, 12747, 15791, 19343, 23791, 28979, 35398, 42887, 52073, 62779, 75804, 90967, 109291, 130605

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

			The a(9) = 1 through a(14) = 11 partitions:
  (54)  (541)  (74)    (543)    (76)      (554)
               (542)   (741)    (544)     (743)
               (5411)  (5421)   (742)     (761)
                       (54111)  (5422)    (5432)
                                (5431)    (5441)
                                (7411)    (7421)
                                (54211)   (54221)
                                (541111)  (54311)
                                          (74111)
                                          (542111)
                                          (5411111)

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The complement is counted by A358909.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358902, A358903, A358908.

Table[Length[Select[IntegerPartitions[n],!GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 49, 53, 64, 81, 113, 128, 131, 151, 161, 165, 169, 225, 243, 256, 311, 343, 361, 512, 719, 729, 1024, 1291, 1321, 1619, 1937, 1957, 2021, 2048, 2093, 2117, 2187, 2197, 2257, 2401, 2805, 2809, 3375, 3671, 4096, 6561

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform, regular, and spans an initial interval of positive integers, so its MM-number 15463 belongs to the sequence.

			The sequence of all uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
    1: {}
    2: {{}}
    3: {{1}}
    4: {{},{}}
    7: {{1,1}}
    8: {{},{},{}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   16: {{},{},{},{}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   32: {{},{},{},{},{}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   64: {{},{},{},{},{},{}}
   81: {{1},{1},{1},{1}}
  113: {{1,2,3}}
  128: {{},{},{},{},{},{},{}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  165: {{1},{2},{3}}
  169: {{1,2},{1,2}}
  225: {{1},{1},{2},{2}}
  243: {{1},{1},{1},{1},{1}}
  256: {{},{},{},{},{},{},{},{}}
  311: {{1,1,1,1,1,1}}
  343: {{1,1},{1,1},{1,1}}
  361: {{1,1,1},{1,1,1}}
  512: {{},{},{},{},{},{},{},{},{}}
  719: {{1,1,1,1,1,1,1}}
  729: {{1},{1},{1},{1},{1},{1}}

Cf. A005176, A007016, A112798, A302242, A306021, A319056, A319189, A320324, A321698, A321717, A322554, A322703, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 13, 15, 16, 17, 29, 31, 32, 33, 41, 43, 47, 51, 55, 59, 64, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 128, 137, 139, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 233, 241, 249, 255, 256, 257, 269, 271

Offset: 1

Gus Wiseman, Dec 14 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
All entries are themselves squarefree numbers (except the powers of 2).
The first odd term not in this sequence but in A302521 is 141, which is the MM-number (see A302242) of {{1},{2,3}}.

			The sequence of all integer partitions whose parts all have the same number of prime factors and whose product of parts is a squarefree number begins: (), (1), (2), (1,1), (3), (1,1,1), (5), (6), (3,2), (1,1,1,1), (7), (10), (11), (1,1,1,1,1), (5,2), (13), (14), (15), (7,2), (5,3), (17), (1,1,1,1,1,1).

Cf. A003963, A005117, A038041, A056239, A073576, A112798, A302242, A302505, A306017, A319056, A319169, A320324, A321717, A321718, A322526, A322528.

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

A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 7, 13, 15, 19, 53, 113, 131, 151, 161, 165, 311, 719, 1291, 1321, 1619, 1937, 1957, 2021, 2093, 2117, 2257, 2805, 3671, 6997, 8161, 10627, 13969, 13987, 14023, 15617, 17719, 17863, 20443, 22207, 22339, 38873, 79349, 84017, 86955, 180503, 202133

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, regular if all vertices appear the same number of times, and strict if there are no repeated parts. For example, {{1,1},{2,3},{2,3}} is uniform and regular but not strict, so its MM-number 15463 does not belong to the sequence. Note that the parts of parts such as {1,1} do not have to be distinct, only the multiset of parts.

			The sequence of all strict uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
      1: {}
      2: {{}}
      3: {{1}}
      7: {{1,1}}
     13: {{1,2}}
     15: {{1},{2}}
     19: {{1,1,1}}
     53: {{1,1,1,1}}
    113: {{1,2,3}}
    131: {{1,1,1,1,1}}
    151: {{1,1,2,2}}
    161: {{1,1},{2,2}}
    165: {{1},{2},{3}}
    311: {{1,1,1,1,1,1}}
    719: {{1,1,1,1,1,1,1}}
   1291: {{1,2,3,4}}
   1321: {{1,1,1,2,2,2}}
   1619: {{1,1,1,1,1,1,1,1}}
   1937: {{1,2},{3,4}}
   1957: {{1,1,1},{2,2,2}}
   2021: {{1,4},{2,3}}
   2093: {{1,1},{1,2},{2,2}}
   2117: {{1,3},{2,4}}
   2257: {{1,1,2},{1,2,2}}
   2805: {{1},{2},{3},{4}}
   3671: {{1,1,1,1,1,1,1,1,1}}
   6997: {{1,1,2,2,3,3}}
   8161: {{1,1,1,1,1,1,1,1,1,1}}
  10627: {{1,1,1,1,2,2,2,2}}
  13969: {{1,2,2},{1,3,3}}
  13987: {{1,1,3},{2,2,3}}
  14023: {{1,1,2},{2,3,3}}
  15617: {{1,1},{2,2},{3,3}}
  17719: {{1,2},{1,3},{2,3}}
  17863: {{1,1,1,1,1,1,1,1,1,1,1}}

Cf. A005117, A007016, A112798, A302242, A306017, A306021, A319056, A319189, A319190, A320324, A321698, A321699, A322554, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322833 Squarefree MM-numbers of strict uniform regular multiset multisystems. Squarefree numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 41, 43, 47, 51, 53, 55, 59, 67, 73, 79, 83, 85, 93, 97, 101, 103, 109, 113, 123, 127, 131, 137, 139, 149, 151, 155, 157, 161, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 227, 233, 241, 249, 255

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, regular if all vertices appear the same number of times, and strict if there are no repeated parts. For example, {{1,2,2},{1,3,3}} is uniform, regular, and strict, so its MM-number 13969 belongs to the sequence. Note that the parts of parts such as {1,2,2} do not have to be distinct, only the multiset of parts.

			The sequence of all strict uniform regular multiset multisystems, together with their MM-numbers, begins:
   1: {}           59: {{7}}         157: {{12}}        269: {{2,8}}
   2: {{}}         67: {{8}}         161: {{1,1},{2,2}} 271: {{1,10}}
   3: {{1}}        73: {{2,4}}       163: {{1,8}}       277: {{17}}
   5: {{2}}        79: {{1,5}}       165: {{1},{2},{3}} 283: {{18}}
   7: {{1,1}}      83: {{9}}         167: {{2,6}}       293: {{1,11}}
  11: {{3}}        85: {{2},{4}}     177: {{1},{7}}     295: {{2},{7}}
  13: {{1,2}}      93: {{1},{5}}     179: {{13}}        311: {{1,1,1,1,1,1}}
  15: {{1},{2}}    97: {{3,3}}       181: {{1,2,4}}     313: {{3,6}}
  17: {{4}}       101: {{1,6}}       187: {{3},{4}}     317: {{1,2,5}}
  19: {{1,1,1}}   103: {{2,2,2}}     191: {{14}}        327: {{1},{10}}
  23: {{2,2}}     109: {{10}}        199: {{1,9}}       331: {{19}}
  29: {{1,3}}     113: {{1,2,3}}     201: {{1},{8}}     335: {{2},{8}}
  31: {{5}}       123: {{1},{6}}     205: {{2},{6}}     341: {{3},{5}}
  33: {{1},{3}}   127: {{11}}        211: {{15}}        347: {{2,9}}
  41: {{6}}       131: {{1,1,1,1,1}} 227: {{4,4}}       349: {{1,3,4}}
  43: {{1,4}}     137: {{2,5}}       233: {{2,7}}       353: {{20}}
  47: {{2,3}}     139: {{1,7}}       241: {{16}}        367: {{21}}
  51: {{1},{4}}   149: {{3,4}}       249: {{1},{9}}     373: {{1,12}}
  53: {{1,1,1,1}} 151: {{1,1,2,2}}   255: {{1},{2},{4}} 381: {{1},{11}}
  55: {{2},{3}}   155: {{2},{5}}     257: {{3,5}}       389: {{4,5}}

Cf. A005117, A007016, A112798, A302242, A306017, A319056, A319189, A320324, A321698, A321699, A322554, A322703.

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

A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

Original entry on oeis.org

1, 15, 33, 51, 55, 85, 91, 93, 123, 155, 161, 165, 177, 187, 201, 203, 205, 249, 255, 295, 299, 301, 327, 329, 335, 341, 377, 381, 415, 451, 465, 471, 511, 527, 537, 545, 553, 559, 561, 573, 611, 615, 633, 635, 649, 667, 679, 697, 703, 707, 723, 737, 785, 831

Offset: 1

Gus Wiseman, Sep 30 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 sequence of terms together with their prime indices begins:
    1: {}
   15: {2,3}
   33: {2,5}
   51: {2,7}
   55: {3,5}
   85: {3,7}
   91: {4,6}
   93: {2,11}
  123: {2,13}
  155: {3,11}
  161: {4,9}
  165: {2,3,5}
  177: {2,17}
  187: {5,7}
  201: {2,19}
  203: {4,10}
  205: {3,13}
  249: {2,23}
  255: {2,3,7}
  295: {3,17}

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

The case including primes and nonsquarefree numbers is A320324.
The version for sum of prime indices is A327901.
The version for mean of prime indices is A327902.
Cf. A001222, A038041, A056239, A078175, A112798, A306017, A306021, A316413, A317583, A322794, A327908.

Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

A327901 Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).

Original entry on oeis.org

1, 35, 143, 209, 247, 391, 493, 629, 667, 851, 901, 1073, 1219, 1333, 1457, 1537, 1891, 1961, 2021, 2201, 2623, 2717, 2759, 2867, 2993, 3053, 3239, 3337, 3827, 3977, 4061, 4183, 4223, 4331, 4387, 4633, 5429, 5633, 5767, 5959, 6157, 6191, 6319, 7081, 7093, 7519

Offset: 1

Gus Wiseman, Sep 30 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 sequence of terms together with their prime indices begins:
     1: {}
    35: {3,4}
   143: {5,6}
   209: {5,8}
   247: {6,8}
   391: {7,9}
   493: {7,10}
   629: {7,12}
   667: {9,10}
   851: {9,12}
   901: {7,16}
  1073: {10,12}
  1219: {9,16}
  1333: {11,14}
  1457: {11,15}
  1537: {10,16}
  1891: {11,18}
  1961: {12,16}
  2021: {14,15}
  2201: {11,20}

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

The version including primes and nonsquarefree numbers is A326534.
The version for number of prime indices is A327900.
The version for mean of prime indices is A327902.
Cf. A001222, A005117, A056239, A112798, A302242, A306021, A320324, A326566, A326574, A327908.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Total/@primeMS/@primeMS[#]&];

cp's OEIS Frontend

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

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A321698 MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

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A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

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A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

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A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

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A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

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A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.

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A322833 Squarefree MM-numbers of strict uniform regular multiset multisystems. Squarefree numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

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A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

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