A370320 -id:A370320 - 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

A368110 Numbers of which it is possible to choose a different divisor of each prime index.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Dec 15 2023

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.

By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - Robert Israel, Feb 15 2024

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   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}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  29: {10}
  30: {1,2,3}

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

Partitions of this type are counted by A239312, complement A370320.
Positions of nonzero terms in A355739.
Complement of A355740.
For just prime divisors we have A368100, complement A355529 (odd A355535).
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.

filter:= proc(n) uses numtheory, GraphTheory; local B,S,F,D,E,G,t,d;
  F:= ifactors(n)[2];
  F:= map(t -> [pi(t[1]),t[2]], F);
  D:= `union`(seq(divisors(t[1]), t = F));
  F:= map(proc(t) local i;seq([t[1],i],i=1..t[2]) end proc,F);
  if nops(D) < nops(F) then return false fi;
  E:= {seq(seq({t,d},d=divisors(t[1])),t = F)};
  S:= map(t -> convert(t,name), [op(F),op(D)]);
  E:= map(e -> map(convert,e,name),E);
  G:= Graph(S,E);
  B:= BipartiteMatching(G);
  B[1] = nops(F);
end proc:
select(filter, [$1..100]); # Robert Israel, Feb 15 2024

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]!={}&]

Heinz numbers of the partitions counted by A239312.

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}]

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}]

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)).

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}]

A371171 Number of integer partitions of n with more parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 9, 12, 18, 26, 34, 50, 65, 92, 121, 161, 209, 274, 353, 456, 590, 745, 950, 1195, 1507, 1885, 2350, 2923, 3611, 4465, 5485, 6735, 8223, 10050, 12195, 14822, 17909, 21653, 26047, 31340, 37557, 44990, 53708, 64068, 76241, 90583, 107418

Offset: 1

Gus Wiseman, Mar 16 2024

nonn

The Heinz numbers of these partitions are given by A370348.

			The partition (3,2,1,1) has 4 parts {1,2,3,4} and 3 distinct divisors of parts {1,2,3}, so is counted under a(7).
The a(0) = 0 through a(8) = 12 partitions:
  .  .  (11)  (111)  (211)   (221)    (222)     (331)      (2222)
                     (1111)  (311)    (2211)    (511)      (3221)
                             (2111)   (3111)    (2221)     (3311)
                             (11111)  (21111)   (3211)     (4211)
                                      (111111)  (4111)     (5111)
                                                (22111)    (22211)
                                                (31111)    (32111)
                                                (211111)   (41111)
                                                (1111111)  (221111)
                                                           (311111)
                                                           (2111111)
                                                           (11111111)

The partitions are ranked by A370348.
The opposite version is A371173, ranked by A371168.
The RHS is represented by A370820, positions of twos A371127.
The version for equality is A371130 (ranks A370802), strict A371128.
For submultisets instead of parts on the LHS we get ranks A371167.
A000005 counts divisors.
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}]

A370595 Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 3, 2, 4, 3, 4, 5, 8, 9, 8, 13, 12, 17, 16, 27, 28, 33, 36, 39, 50, 58, 65, 75, 93, 94, 112, 125, 148, 170, 190, 209, 250, 273, 305, 341, 403, 432, 484, 561, 623, 708, 765, 873, 977, 1109, 1178, 1367, 1493, 1669, 1824, 2054, 2265, 2521, 2770

Offset: 0

Gus Wiseman, Mar 03 2024

nonn

For example, the only choice for the partition (9,9,6,6,6) is {1,2,3,6,9}.

			The a(1) = 1 through a(15) = 13 partitions (A = 10, B = 11, C = 12, D = 13):
  1  .  21  22  .  33   322  71   441  55    533   B1    553   77    933
            31     51   421  332  522  442   722   444   733   D1    B22
                   321       422  531  721   731   552   751   B21   B31
                             521       4321  4322  4332  931   4433  4443
                                             5321  4431  4432  5441  5442
                                                   5322  5332  6332  5532
                                                   5421  5422  7322  6621
                                                   6321  6322  7421  7332
                                                         7321        7422
                                                                     7521
                                                                     8421
                                                                     9321
                                                                     54321

For no choices we have A370320, complement A239312.
The version for prime factors (not all divisors) is A370594, ranks A370647.
For multiple choices we have A370803, ranks A370811.
These partitions have ranks A370810.
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.
A355741, A355744, A355745 choose prime factors of prime indices.
A370592 counts partitions with choosable prime factors, ranks A368100.
A370593 counts partitions without choosable prime factors, ranks A355529.
A370804 counts non-condensed partitions with no ones, complement A370805.
A370814 counts factorizations with choosable divisors, complement A370813.
Cf. A355535, A355739, A355740, A367867, A367904, A368110, A370583, A370584, A370806, A370808.

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

More terms from Jinyuan Wang, Feb 14 2025

cp's OEIS Frontend

A239312 Number of condensed integer partitions of n.

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A368110 Numbers of which it is possible to choose a different divisor of each prime index.

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

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

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

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

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