A370814 -id:A370814 - 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

A368413 Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 2, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 1, 10, 0, 0, 0, 1, 0, 0, 0, 10, 0, 0, 1, 1, 0, 0, 0, 7, 4, 0, 0, 2, 0, 0

Offset: 1

Gus Wiseman, Dec 27 2023

nonn

For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is counted under a(36).

			The a(1) = 0 through a(24) = 3 factorizations:
 ... 2*2 ... 2*4   3*3 .. 2*2*3 ... 2*8     . 2*3*3 . 2*2*5 ... 2*2*6
             2*2*2                  4*4                         2*3*4
                                    2*2*4                       2*2*2*3
                                    2*2*2*2

For unlabeled graphs: A140637, complement A134964.
For labeled graphs: A367867, A367868, A140638, complement A133686.
For set-systems: A367903, ranks A367907, complement A367902, ranks A367906.
For non-isomorphic set-systems: A368094, A368409, complement A368095.
For non-isomorphic multiset partitions: A368097, A355529, A368411.
Complement for non-isomorphic multiset partitions: A368098, A368100.
The complement is counted by A368414.
For non-isomorphic set multipartitions: A368421, complement A368422.
For divisors instead of prime factors: A370813, complement A370814.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A340596, A340653, A367769, A367901, A368187, A368412.

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], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]=={}&]],{n,100}]

a(n) + A368414(n) = A001055(n).

A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is not counted under a(36).

			The a(n) factorizations for selected n:
  1    6      12     24      30       60        72      120
       2*3    2*6    2*12    2*15     2*30      2*36    2*60
              3*4    3*8     3*10     3*20      3*24    3*40
                     4*6     5*6      4*15      4*18    4*30
                             2*3*5    5*12      6*12    5*24
                                      6*10      8*9     6*20
                                      2*3*10            8*15
                                      2*5*6             10*12
                                      3*4*5             2*3*20
                                                        2*5*12
                                                        2*6*10
                                                        3*4*10
                                                        3*5*8
                                                        4*5*6

For labeled graphs: A133686, complement A367867, A367868, A140638.
For unlabeled graphs: A134964, complement A140637.
For set-systems: A367902, ranks A367906, complement A367903, ranks A367907.
For non-isomorphic set-systems: A368095, complement A368094, A368409.
Complementary non-isomorphic multiset partitions: A368097, A355529, A368411.
For non-isomorphic multiset partitions: A368098, A368100.
The complement is counted by A368413.
For non-isomorphic set multipartitions: A368422, complement A368421.
For divisors instead of prime factors: A370813, complement A370814.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A340596, A340653, A367769, A367901, A368187, A368412, A368413.

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], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]!={}&]],{n,100}]

a(n) = A001055(n) - A368413(n).

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

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 39, 61, 122, 203, 315, 469, 676, 952, 1307, 1771, 3542, 5708, 8432, 11877, 16123, 21415, 27835, 35757, 45343, 57010, 70778, 87384, 106479, 129304, 155802, 187223, 374446, 588130, 835800, 1124981, 1456282, 1841361, 2281772, 2791896, 3367162

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {1,3,4}
                         {2,3,4}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A134964, complement A140637.
Simple graphs not of this type are counted by A367867, covering A367868.
Set systems of this type are counted by A367902, ranks A367906.
Set systems not of this type are counted by A367903, ranks A367907.
Set systems uniquely of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
Factorizations are counted by A368414/A370814, complement A368413/A370813.
For prime indices we have A370582, differences A370586.
The complement for prime indices is A370583, differences A370587.
The complement is A370637, differences A370589, without ones A370643.
The case of a unique choice is A370638, maxima A370640, differences A370641.
First differences are A370639.
The minimal case of the complement is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]

a(2^n - 1) = A367902(n).

Partial sums of A370639.

a(19)-a(40) from Alois P. Heinz, Mar 09 2024

A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 6, 9, 10, 22, 25, 28, 30, 34, 42, 45, 62, 63, 66, 75, 82, 92, 98, 99, 102, 104, 110, 118, 121, 134, 140, 147, 152, 153, 156, 166, 170, 186, 210, 218, 228, 230, 232, 234, 246, 254, 260, 275, 276, 279, 289, 308, 310, 314, 315, 330, 342, 343, 344, 348, 350

Offset: 1

Gus Wiseman, Mar 14 2024

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.

All squarefree terms are even.

			The prime indices of 1617 are {2,4,4,5}, with distinct divisors {1,2,4,5}, so 1617 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   42: {1,2,4}
   45: {2,2,3}
   62: {1,11}
   63: {2,2,4}
   66: {1,2,5}
   75: {2,3,3}
   82: {1,13}
   92: {1,1,9}
   98: {1,4,4}
   99: {2,2,5}
  102: {1,2,7}
  104: {1,1,1,6}

For factors instead of divisors on the RHS we have A319899.
A version for binary indices is A367917.
For (greater than) instead of (equal) we have A370348, counted by A371171.
The RHS is A370820, for prime factors instead of divisors A303975.
Partitions of this type are counted by A371130, strict A371128.
For divisors instead of factors on LHS we have A371165, counted by A371172.
For only distinct prime factors on LHS we have A371177, counted by A371178.
Other inequalities: A371166, A371167, A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.

Select[Range[100],PrimeOmega[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A001222(a(n)) = A370820(a(n)).

A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 25, 67, 134, 309, 709, 1579, 3420, 7240, 15077, 30997, 61994, 125364, 253712, 512411, 1032453, 2075737, 4166469, 8352851, 16731873, 33497422, 67038086, 134130344, 268328977, 536741608, 1073586022, 2147296425, 4294592850, 8589346462, 17179033384

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370582, differences A370586.
For prime indices we have A370583, differences A370587.
First differences are A370589.
The complement is counted by A370636, differences A370639.
The case without ones is A370643.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The minimal case is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A072639, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

a(2^n - 1) = A367903(n).

Partial sums of A370589.

a(21)-a(34) from Alois P. Heinz, Mar 09 2024

A370803 Number of integer partitions of n such that more than one set can be obtained by choosing a different divisor of each part.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 10, 11, 15, 18, 25, 28, 39, 45, 59, 66, 83, 101, 123, 150, 176, 213, 252, 301, 352, 426, 497, 589, 684, 802, 939, 1095, 1270, 1480, 1718, 1985, 2289, 2645, 3056, 3489, 4019, 4590, 5289, 6014, 6877, 7817, 8955, 10134, 11551, 13085

Offset: 0

Gus Wiseman, Mar 03 2024

nonn

			The partition (6,4,4,1) has two choices, namely {1,2,4,6} and {1,2,3,4}, so is counted under a(15).
The a(0) = 0 through a(13) = 18 partitions (A..D = 10..13):
  .  .  2   3   4   5    6    7    8     9     A     B     C     D
                    32   42   43   44    54    64    65    66    76
                    41        52   53    63    73    74    75    85
                              61   62    72    82    83    84    94
                                   431   81    91    92    93    A3
                                         432   433   A1    A2    B2
                                         621   532   443   543   C1
                                               541   542   633   544
                                               622   632   642   643
                                               631   641   651   652
                                                     821   732   661
                                                           741   742
                                                           822   832
                                                           831   841
                                                           921   922
                                                                 A21
                                                                 5431
                                                                 6421

Including partitions with one choice gives A239312, complement A370320.
For a unique choice we have A370595, ranks A370810.
These partitions have ranks A370811.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355733 counts divisor-choices of prime indices.
A355741, A355744, A355745 choose prime factors of prime indices.
A370592 counts factor-choosable partitions, ranks A368100.
A370593 counts non-factor-choosable partitions, ranks A355529.
Cf. A355739, A368110, A370594, A370804, A370805, A370808, A370814.

Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]>1&]],{n,0,30}]

a(n) = A239312(n) - A370595(n). - Jinyuan Wang, Feb 14 2025

More terms from Jinyuan Wang, Feb 14 2025

A371130 Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 4, 2, 4, 5, 5, 11, 10, 16, 17, 21, 26, 32, 44, 53, 69, 71, 101, 110, 148, 168, 205, 249, 289, 356, 418, 502, 589, 716, 812, 999, 1137, 1365, 1566, 1873, 2158, 2537, 2942, 3449, 4001, 4613, 5380, 6193, 7220, 8224, 9575, 10926, 12683, 14430

Offset: 0

Gus Wiseman, Mar 17 2024

nonn

The Heinz numbers of these partitions are given by A370802.

			The partition (6,2,2,1) has 4 parts and 4 distinct divisors of parts {1,2,3,6} so is counted under a(11).
The a(1) = 1 through a(11) = 11 partitions:
  (1)  .  (21)  (22)  .  (33)   (322)  (71)   (441)   (55)    (533)
                (31)     (51)   (421)  (332)  (522)   (442)   (722)
                         (321)         (422)  (531)   (721)   (731)
                         (411)         (521)  (4311)  (4321)  (911)
                                              (6111)  (6211)  (4322)
                                                              (4331)
                                                              (5321)
                                                              (5411)
                                                              (6221)
                                                              (6311)
                                                              (8111)

The LHS is represented by A001222, distinct A000021.
These partitions are ranked by A370802.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
For (greater than) instead of (equal to) we have A371171, ranks A370348.
For submultisets instead of parts on the LHS we have A371172.
For (less than) instead of (equal to) we have A371173, ranked by A371168.
Counting only distinct parts on the LHS gives A371178, ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

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

cp's OEIS Frontend

A239312 Number of condensed integer partitions of n.

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A368413 Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.

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A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

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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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A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

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A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

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