A370592 -id:A370592 - cp's OEIS frontend

A239312 Number of condensed integer partitions of n.

1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044

Offset: 0

Clark Kimberling, Mar 15 2014

nonn

Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.

Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024

			a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
From _Gus Wiseman_, Mar 12 2024: (Start)
The a(1) = 1 through a(9) = 10 condensed partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                   (3,1)  (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,2,2)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,2,2)  (4,4,1)
                                                   (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..100 (first 84 terms from Manfred Scheucher)
Manfred Scheucher, Python Script

The strict case is A000009.
These partitions have ranks A368110, complement A355740.
The complement is counted by A370320.
The version for prime factors (not all divisors) is A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
For a unique choice we have A370595, ranks A370810.
For multiple choices we have A370803, ranks A370811.
The case without ones is A370805, complement A370804.
The version for factorizations is A370814, complement A370813.
A000005 counts divisors.
A000041 counts integer partitions.
A237685 counts partitions of depth 1, or A353837 if we include depth 0.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.

b:= proc(n,i) option remember; `if`(n=0, {[]},
      `if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
       sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> nops(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 01 2019

u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0,   30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]], {n,0,30}] (* Gus Wiseman, Mar 12 2024 *)

Typo in definition corrected by Manfred Scheucher, May 29 2015

Name edited by Gus Wiseman, Mar 13 2024

A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 10, 12, 19, 26, 38, 51, 71, 94, 126, 165, 219, 285, 369, 472, 605, 766, 973, 1226, 1538, 1917, 2387, 2955, 3657, 4497, 5532, 6754, 8251, 10033, 12190, 14748, 17831, 21471, 25825, 30976, 37111, 44331, 52897, 62952, 74829, 88755, 105145, 124307

Offset: 0

Gus Wiseman, Feb 29 2024

nonn

			The a(0) = 0 through a(7) = 12 partitions:
  .  (1)  (11)  (21)   (22)    (41)     (33)      (61)
                (111)  (31)    (221)    (42)      (322)
                       (211)   (311)    (51)      (331)
                       (1111)  (2111)   (222)     (421)
                               (11111)  (321)     (511)
                                        (411)     (2221)
                                        (2211)    (3211)
                                        (3111)    (4111)
                                        (21111)   (22111)
                                        (111111)  (31111)
                                                  (211111)
                                                  (1111111)

The complement for divisors instead of factors is A239312, ranks A368110.
These partitions have ranks A355529, complement A368100.
The complement for set-systems is A367902, ranks A367906, unlabeled A368095.
The version for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368097, complement A368098.
The version for factorizations is A368413, complement A368414.
The complement is counted by A370592.
For a unique choice we have A370594, ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A367771, A367867, A367905, A370583, A370585, A370586, A370636.

Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]],{n,0,30}]

a(n) = A000041(n) - A370592(n).

A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 10, 11, 14, 17, 19, 23, 29, 30, 39, 41, 51, 58, 66, 78, 82, 102, 110, 132, 144, 162, 186, 210, 228, 260, 296, 328, 366, 412, 462, 512, 560, 638, 692, 764, 860, 924, 1028, 1122, 1276, 1406, 1528, 1721, 1898, 2056, 2318, 2506, 2812, 3020, 3442

Offset: 0

Gus Wiseman, Mar 05 2024

nonn

			For the partitions of 5 we have the following choices:
      (5): {{1},{5}}
     (41): {{1,1},{1,2},{1,4}}
     (32): {{1,1},{1,2},{1,3},{2,3}}
    (311): {{1,1,1},{1,1,3}}
    (221): {{1,1,1},{1,1,2},{1,2,2}}
   (2111): {{1,1,1,1},{1,1,1,2}}
  (11111): {{1,1,1,1,1}}
So a(5) = 4.

For just prime factors we have A370809.
The version for factorizations is A370816, for just prime factors A370817.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts condensed partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355733 counts choices of divisors of prime indicec.
A370320 counts non-condensed partitions, ranks A355740.
A370592 counts factor-choosable partitions, complement A370593.
Cf. A000792, A048249, A066739, A319055, A355737, A355739, A370348, A370595, A370803, A370810.

Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@IntegerPartitions[n]],{n,0,30}]

Terms a(31) onward from Max Alekseyev, Sep 17 2024

A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 04 2024

nonn

A multiset is condensed iff it is possible to choose a different divisor of each element.

			The a(96) = 4 factorizations: (2*2*2*2*2*3), (2*2*2*2*6), (2*2*2*3*4), (2*2*2*12).

Partitions not of this type are counted by A239312, ranks A368110.
Factors instead of divisors: A368413, complement A368414, unique A370645.
Partitions of this type are counted by A370320, ranks A355740.
Subsets of this type: A370583 and A370637, complement A370582 and A370636.
The complement is counted by A370814, partitions A370592, ranks A368100.
For a unique choice we have A370815, partitions A370595, ranks A370810.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A340596, A340653, A355529, A355739, A355741, A370804, A370807.

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],Length[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]==0&]],{n,100}]

A370320 Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 9, 13, 20, 28, 40, 54, 74, 102, 135, 180, 235, 310, 397, 516, 658, 843, 1066, 1349, 1687, 2119, 2634, 3273, 4045, 4995, 6128, 7517, 9171, 11181, 13579, 16457, 19884, 23992, 28859, 34646, 41506, 49634, 59211, 70533, 83836, 99504, 117867

Offset: 0

Gus Wiseman, Mar 02 2024

nonn

Includes all partitions containing 1.

			The a(0) = 0 through a(8) = 13 partitions:
  .  .  (11)  (111)  (211)   (221)    (222)     (331)      (611)
                     (1111)  (311)    (411)     (511)      (2222)
                             (2111)   (2211)    (2221)     (3221)
                             (11111)  (3111)    (3211)     (3311)
                                      (21111)   (4111)     (4211)
                                      (111111)  (22111)    (5111)
                                                (31111)    (22211)
                                                (211111)   (32111)
                                                (1111111)  (41111)
                                                           (221111)
                                                           (311111)
                                                           (2111111)
                                                           (11111111)

The complement is counted by A239312 (condensed partitions).
These partitions have ranks A355740.
Factorizations in the case of prime factors are A368413, complement A368414.
The complement for prime factors is A370592, ranks A368100.
The version for prime factors (not all divisors) is A370593, ranks A355529.
For a unique choice we have A370595, ranks A370810.
For multiple choices we have A370803, ranks A370811.
The case without ones is A370804, complement A370805.
The version for factorizations is A370813, complement A370814.
A000005 counts divisors.
A000041 counts integer partitions.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A355535, A355739, A367867, A368097, A368110, A370583, A370584, A370594, A370806, A370807, A370808.

Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]==0&]],{n,0,30}]

a(31)-a(47) from Alois P. Heinz, Mar 03 2024

A370814 Number of condensed integer factorizations of n into unordered factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Mar 04 2024

nonn

A multiset is condensed iff it is possible to choose a different divisor of each element.

			The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).

Partitions of this type are counted by A239312, ranks A368110.
Factors instead of divisors: A368414, complement A368413, unique A370645.
Partitions not of this type are counted by A370320, ranks A355740.
Subsets of this type: A370582 and A370636, complement A370583 and A370637.
The complement is counted by A370813, partitions A370593, ranks A355529.
For a unique choice we have A370815, partitions A370595, ranks A370810.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A340596, A340653, A355739, A370592, A370803, A370804, A370805.

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],Length[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]>0&]],{n,100}]

A370583 Number of subsets of {1..n} such that it is not possible to choose a different prime factor of each element.

Original entry on oeis.org

0, 1, 2, 4, 10, 20, 44, 88, 204, 440, 908, 1816, 3776, 7552, 15364, 31240, 63744, 127488, 257592, 515184, 1036336, 2079312, 4166408, 8332816, 16709632, 33470464, 66978208, 134067488, 268236928, 536473856, 1073233840, 2146467680, 4293851680, 8588355424, 17177430640

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

			The a(0) = 0 through a(5) = 20 subsets:
  .  {1}  {1}    {1}      {1}        {1}
          {1,2}  {1,2}    {1,2}      {1,2}
                 {1,3}    {1,3}      {1,3}
                 {1,2,3}  {1,4}      {1,4}
                          {2,4}      {1,5}
                          {1,2,3}    {2,4}
                          {1,2,4}    {1,2,3}
                          {1,3,4}    {1,2,4}
                          {2,3,4}    {1,2,5}
                          {1,2,3,4}  {1,3,4}
                                     {1,3,5}
                                     {1,4,5}
                                     {2,3,4}
                                     {2,4,5}
                                     {1,2,3,4}
                                     {1,2,3,5}
                                     {1,2,4,5}
                                     {1,3,4,5}
                                     {2,3,4,5}
                                     {1,2,3,4,5}

Multisets of this type are ranked by A355529, complement A368100.
For divisors instead of factors we have A355740, complement A368110.
The complement for set-systems is A367902, ranks A367906, unlabeled A368095.
The version for set-systems is A367903, ranks A367907, unlabeled A368094.
For non-isomorphic multiset partitions we have A368097, complement A368098.
The version for factorizations is A368413, complement A368414.
The complement is counted by A370582.
For a unique choice we have A370584.
Partial sums of A370587, complement A370586.
The minimal case is A370591.
The version for partitions is A370593, complement A370592.
For binary indices instead of factors we have A370637, complement A370636.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A001055, A001414, A003963, A005117, A045778, A355739, A355745, A367867, A367905, A368187.

Table[Length[Select[Subsets[Range[n]], Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]],{n,0,10}]

a(n) = 2^n - A370582(n).

a(19)-a(34) from Alois P. Heinz, Feb 27 2024

A370585 Number of maximal subsets of {1..n} such that it is possible to choose a different prime factor of each element.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 5, 7, 11, 25, 25, 38, 38, 84, 150, 178, 178, 235, 235, 341, 579, 1235, 1235, 1523, 1968, 4160, 4824, 6840, 6840, 9140, 9140, 10028, 16264, 33956, 48680, 56000, 56000, 116472, 186724, 223884, 223884, 290312, 290312, 403484, 484028, 1001420

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

First differs from A307984 at a(21) = 579, A307984(21) = 578. The difference is due to the set {10,11,13,14,15,17,19,21}, which is not a basis because log(10) + log(21) = log(14) + log(15).

Also length-pi(n) subsets of {1..n} such that it is possible to choose a different prime factor of each element.

			The a(0) = 1 through a(8) = 7 subsets:
  {}  {}  {2}  {2,3}  {2,3}  {2,3,5}  {2,3,5}  {2,3,5,7}  {2,3,5,7}
                      {3,4}  {3,4,5}  {2,5,6}  {2,5,6,7}  {2,5,6,7}
                                      {3,4,5}  {3,4,5,7}  {3,4,5,7}
                                      {3,5,6}  {3,5,6,7}  {3,5,6,7}
                                      {4,5,6}  {4,5,6,7}  {3,5,7,8}
                                                          {4,5,6,7}
                                                          {5,6,7,8}

Multisets of this type are ranked by A368100, complement A355529.
Factorizations of this type are counted by A368414, complement A368413.
The version for set-systems is A368601, max of A367902 (complement A367903).
This is the maximal case of A370582, complement A370583, cf. A370584.
A different kind of maximality is A370586, complement A370587.
The case containing n is A370590, complement A370591.
Partitions of this type (choosable) are A370592, complement A370593.
For binary indices instead of factors we have A370640, cf. A370636, A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A005117, A045778, A133686, A333331, A355739, A355740, A355744, A355745, A367905, A368110.

Table[Length[Select[Subsets[Range[n], {PrimePi[n]}],Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,10}]

More terms from Jinyuan Wang, Feb 14 2025

A370582 Number of subsets of {1..n} such that it is possible to choose a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 20, 40, 52, 72, 116, 232, 320, 640, 1020, 1528, 1792, 3584, 4552, 9104, 12240, 17840, 27896, 55792, 67584, 83968, 130656, 150240, 198528, 397056, 507984, 1015968, 1115616, 1579168, 2438544, 3259680, 3730368, 7460736, 11494656, 16145952, 19078464, 38156928

Offset: 0

Gus Wiseman, Feb 25 2024

nonn

			The a(0) = 1 through a(6) = 20 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {6}
                             {2,5}    {2,3}
                             {3,4}    {2,5}
                             {3,5}    {2,6}
                             {4,5}    {3,4}
                             {2,3,5}  {3,5}
                             {3,4,5}  {3,6}
                                      {4,5}
                                      {4,6}
                                      {5,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

The version for set-systems is A367902, ranks A367906, unlabeled A368095.
The complement for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368098, complement A368097.
Multisets of this type are ranked by A368100, complement A355529.
For divisors instead of factors we have A368110, complement A355740.
The version for factorizations is A368414, complement A368413.
The complement is counted by A370583.
For a unique choice we have A370584.
The maximal case is A370585.
Partial sums of A370586, complement A370587.
The version for partitions is A370592, complement A370593.
For binary indices instead of factors we have A370636, complement A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A001055, A001414, A003963, A005117, A045778, A133686, A355739, A355744, A355745, A367905.

Table[Length[Select[Subsets[Range[n]],Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]>0&]],{n,0,10}]

a(p) = 2 * a(p-1) for prime p. - David A. Corneth, Feb 25 2024

a(n) = 2^n - A370583(n).

a(19) from David A. Corneth, Feb 25 2024

a(20)-a(41) from Alois P. Heinz, Feb 25 2024

A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 18, 36, 48, 68, 104, 208, 284, 568, 888, 1296, 1548, 3096, 3968, 7936, 10736, 15440, 24008, 48016, 58848, 73680, 114368, 132608, 176240, 352480, 449824, 899648, 994976, 1399968, 2160720, 2859584, 3296048, 6592096, 10156672, 14214576, 16892352

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3).

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {2,3}
                             {2,5}    {2,5}
                             {3,4}    {2,6}
                             {3,5}    {3,4}
                             {4,5}    {3,5}
                             {2,3,5}  {3,6}
                             {3,4,5}  {4,5}
                                      {4,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

For divisors instead of factors we have A051026, cf. A368110, A355740.
The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
For existence we have A370582, differences A370586.
For nonexistence we have A370583, differences A370587.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, cf. A370592, A370593.
For binary indices instead of factors we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
For unlabeled multiset partitions we have A370646, cf. A368098, A368097.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000720, A003963, A005117, A045778, A133686, A307984, A355739, A355744, A355745, A367905.

Table[Length[Select[Subsets[Range[n]], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 28 2025

cp's OEIS Frontend

A239312 Number of condensed integer partitions of n.

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A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.

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A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

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A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.

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A370320 Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.

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A370814 Number of condensed integer factorizations of n into unordered factors > 1.

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A370583 Number of subsets of {1..n} such that it is not possible to choose a different prime factor of each element.

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A370585 Number of maximal subsets of {1..n} such that it is possible to choose a different prime factor of each element.

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A370582 Number of subsets of {1..n} such that it is possible to choose a different prime factor of each element.