A370592 -id:A370592 - cp's OEIS frontend

A387137 Number of integer partitions of n whose parts do not have choosable sets of strict integer partitions.

0, 0, 1, 1, 3, 4, 6, 9, 14, 20, 29, 39, 56, 74, 101, 134, 178, 232, 305, 392, 508, 646, 825, 1042, 1317, 1649, 2066, 2567, 3190, 3937, 4859, 5960, 7306, 8914, 10863, 13183, 15984, 19304, 23288, 28003, 33631, 40272, 48166, 57453, 68448, 81352, 96568, 114383

Offset: 0

Gus Wiseman, Sep 02 2025

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
a(n) is the number of integer partitions of n such that it is not possible to choose a sequence of distinct strict integer partitions, one of each part.
Also the number of integer partitions of n with at least one part k whose multiplicity exceeds A000009(k).

			The a(2) = 1 through a(8) = 14 partitions:
  (11)  (111)  (22)    (221)    (222)     (322)      (422)
               (211)   (311)    (411)     (511)      (611)
               (1111)  (2111)   (2211)    (2221)     (2222)
                       (11111)  (3111)    (3211)     (3221)
                                (21111)   (4111)     (3311)
                                (111111)  (22111)    (4211)
                                          (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement for initial intervals is A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
For divisors instead of strict partitions we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of strict partitions we have A370593, ranks A355529.
For initial intervals instead of strict partitions we have A387118, ranks A387113.
For all partitions instead of strict partitions we have A387134, ranks A387577.
These partitions are ranked by A387176.
The complement is counted by A387178, ranks A387177.
The complement for partitions is A387328, ranks A387576.
The version for constant partitions is A387329, ranks A387180.
The complement for constant partitions is A387330, ranks A387181.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A355535, A367867, A367901, A367905, A383706, A387115.

strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[strptns/@#],UnsameQ@@#&]]==0&]],{n,0,15}]

A370806 Number of non-strict condensed integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 3, 2, 4, 4, 8, 9, 11, 14, 19, 24, 29, 39, 47, 58, 70, 85, 104, 129, 152, 184, 223, 264, 313, 374, 442, 524, 617, 719, 852, 993, 1159, 1344, 1579, 1817, 2114, 2440, 2826, 3250, 3750, 4297, 4944, 5662, 6475, 7404, 8462, 9634, 10972, 12480

Offset: 0

Gus Wiseman, Mar 04 2024

nonn

These are non-strict partitions such that it is possible to choose a different divisor of each part.

			The a(4) = 1 through a(13) = 9 partitions:
  (22)  .  (33)  (322)  (44)   (441)  (55)   (443)   (66)    (544)
                        (332)  (522)  (433)  (533)   (444)   (553)
                        (422)         (442)  (722)   (552)   (661)
                                      (622)  (4322)  (633)   (733)
                                                     (822)   (922)
                                                     (4332)  (4432)
                                                     (4431)  (5332)
                                                     (5322)  (5422)
                                                             (6322)

This is the non-strict case of A239312, complement A370320.
These partitions have as ranks the nonsquarefree terms of A368110.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370592 counts factor-choosable partitions, complement A370593.
A370814 counts condensed factorizations, complement A370813.
Cf. A355739, A355740, A370595, A370804, A370808, A370810.

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

More terms from Jinyuan Wang, Feb 14 2025

A370807 Number of integer partitions of n into parts > 1 such that it is not possible to choose a different prime factor of each part.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 1, 4, 4, 8, 9, 15, 17, 25, 30, 43, 54, 72, 87, 115, 139, 181, 224, 283, 342, 429, 519, 647, 779, 967

Offset: 0

Gus Wiseman, Mar 04 2024

			The a(0) = 0 through a(11) = 9 partitions:
  .  .  .  .  (22)  .  (33)   (322)  (44)    (333)   (55)     (443)
                       (42)          (332)   (432)   (82)     (533)
                       (222)         (422)   (522)   (433)    (542)
                                     (2222)  (3222)  (442)    (632)
                                                     (622)    (722)
                                                     (3322)   (3332)
                                                     (4222)   (4322)
                                                     (22222)  (5222)
                                                              (32222)

These partitions are ranked by the odd terms of A355529, complement A368100.
The version for set-systems is A367903, complement A367902.
The version for factorizations is A368413, complement A368414.
With ones allowed we have A370593, complement A370592.
For a unique choice we have A370594, ranks A370647.
The version for divisors instead of factors is A370804, complement A370805.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts condensed partitions, ranks A368110.
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],FreeQ[#,1] && Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]==0&]],{n,0,30}]

A370811 Numbers such that more than one set can be obtained by choosing a different divisor of each prime index.

Original entry on oeis.org

3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115, 117, 119

Offset: 1

Gus Wiseman, Mar 13 2024

nonn

A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.

			The prime indices of 70 are {1,3,4}, with choices (1,3,4) and (1,3,2), so 70 is in the sequence.
The terms together with their prime indices begin:
     3: {2}      43: {14}        79: {22}       115: {3,9}
     5: {3}      46: {1,9}       83: {23}       117: {2,2,6}
     7: {4}      47: {15}        85: {3,7}      119: {4,7}
    11: {5}      49: {4,4}       86: {1,14}     122: {1,18}
    13: {6}      51: {2,7}       87: {2,10}     123: {2,13}
    14: {1,4}    53: {16}        89: {24}       127: {31}
    15: {2,3}    55: {3,5}       91: {4,6}      129: {2,14}
    17: {7}      57: {2,8}       93: {2,11}     130: {1,3,6}
    19: {8}      58: {1,10}      94: {1,15}     131: {32}
    21: {2,4}    59: {17}        95: {3,8}      133: {4,8}
    23: {9}      61: {18}        97: {25}       137: {33}
    26: {1,6}    65: {3,6}      101: {26}       138: {1,2,9}
    29: {10}     67: {19}       103: {27}       139: {34}
    31: {11}     69: {2,9}      105: {2,3,4}    141: {2,15}
    33: {2,5}    70: {1,3,4}    106: {1,16}     142: {1,20}
    35: {3,4}    71: {20}       107: {28}       143: {5,6}
    37: {12}     73: {21}       109: {29}       145: {3,10}
    38: {1,8}    74: {1,12}     111: {2,12}     146: {1,21}
    39: {2,6}    77: {4,5}      113: {30}       149: {35}
    41: {13}     78: {1,2,6}    114: {1,2,8}    151: {36}

For no choices we have A355740, counted by A370320.
For at least one choice we have A368110, counted by A239312.
Partitions of this type are counted by A370803.
For a unique choice we have A370810, counted by A370595 and A370815.
A000005 counts divisors.
A006530 gives greatest prime factor, least A020639.
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.
A370814 counts factorizations with choosable divisors, complement A370813.
Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370594, A370647, A370808, A370816.

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

A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 25, 34, 49, 65, 89, 118, 158, 206, 271, 349, 453, 578, 740, 935, 1186, 1486, 1865, 2322, 2890, 3572, 4415, 5423, 6659, 8134, 9927, 12062, 14643, 17706, 21387, 25746, 30957, 37109, 44433, 53054, 63273, 75276, 89444, 106044

Offset: 0

Gus Wiseman, Aug 29 2025

Number of integer partitions of n such that it is not possible to choose a sequence of distinct integer partitions, one of each part.

Also the number of integer partitions of n with at least one part k satisfying that the multiplicity of k exceeds the number of integer partitions of k.

			The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (211)   (311)    (222)     (511)      (611)
               (1111)  (2111)   (411)     (2221)     (2222)
                       (11111)  (2211)    (3211)     (3311)
                                (3111)    (4111)     (4211)
                                (21111)   (22111)    (5111)
                                (111111)  (31111)    (22211)
                                          (211111)   (32111)
                                          (1111111)  (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

These partitions are ranked by A276079.
For divisors instead of partitions we have A370320, complement A239312.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of partitions we have A370593, ranks A355529.
For initial intervals instead of partitions we have A387118, complement A238873.
For just choices of strict partitions we have A387137.
The complement is counted by A387328, ranks A276078.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
Cf. A335433, A355535, A367867, A367901, A367903, A367905, A367907, A370583, A370594.

Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]]==0&]],{n,0,15}]

A370815 Number of integer factorizations of n into unordered factors > 1, such that only one set can be obtained by choosing a different divisor of each factor.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 06 2024

nonn

			The a(432) = 3 factorizations: (2*2*3*4*9), (2*3*3*4*6), (2*6*6*6).

For partitions and prime factors we have A370594, ranks A370647.
Partitions of this type are counted by A370595, ranks A370810.
For prime factors we have A370645, subsets A370584.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A239312 counts condensed partitions, ranks A355740, complement A370320.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A368414 counts factor-choosable factorizations, complement A368413.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A340596, A340653, A355529, A355739, A368110, A370592, A370638, A370803.

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[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,100}]

A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Aug 24 2025

The initial interval of a nonnegative integer x is the set {1,...,x}.
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.
We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1,2,3},{1},{1,3},{2}) is not.
This sequence lists all numbers k such that if the prime indices of k are (x1,x2,...,xz), then the sequence of sets (initial intervals) ({1,...,x1},{1,...,x2},...,{1,...,xz}) is not choosable.

			The prime indices of 18 are {1,2,2}, with initial intervals ({1},{1,2},{1,2}), which have choices (1,1,1), (1,1,2), (1,2,1), (1,2,2), and since none of these are strict, 18 is in the sequence.
The prime indices of 85 are {3,7}, with initial intervals {{1,2,3},{1,2,3,4,5,6,7}}, which are choosable, so 85 is in not the sequence.
The prime indices of 90 are {1,2,2,3}, with initial intervals {{1},{1,2},{1,2},{1,2,3}}, which are not choosable, so 90 is in the sequence.
The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

For partitions instead of initial intervals we have A276079, complement A276078.
For prime factors instead of initial intervals we have A355529, complement A368100.
For divisors instead of initial intervals we have A355740, complement A368110.
These are the positions of 0 in A387111, complement A387134.
The complement is A387112.
Partitions of this type are counted by A387118, complement A238873.
For strict partitions instead of initial intervals we have A387137, complement A387176.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
A370585 counts maximal subsets with choosable prime factors.
Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A355747, A370592, A383706, A387110.

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

A370588 Number of subsets of {1..n} containing n such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

0, 0, 1, 2, 2, 6, 6, 18, 12, 20, 36, 104, 76, 284, 320, 408, 252, 1548, 872, 3968, 2800, 4704, 8568, 24008, 10832, 14832, 40688, 18240, 43632, 176240, 97344, 449824, 95328, 404992, 760752, 698864, 436464, 3296048, 3564576, 4057904, 2677776, 16892352, 8676576

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

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

			The a(0) = 0 through a(8) = 12 subsets:
  .  .  {2}  {3}    {4}    {5}      {2,6}    {7}        {8}
             {2,3}  {3,4}  {2,5}    {3,6}    {2,7}      {3,8}
                           {3,5}    {4,6}    {3,7}      {5,8}
                           {4,5}    {2,5,6}  {4,7}      {6,8}
                           {2,3,5}  {3,5,6}  {5,7}      {7,8}
                           {3,4,5}  {4,5,6}  {2,3,7}    {3,5,8}
                                             {2,5,7}    {3,7,8}
                                             {2,6,7}    {5,6,8}
                                             {3,4,7}    {5,7,8}
                                             {3,5,7}    {6,7,8}
                                             {3,6,7}    {3,5,7,8}
                                             {4,5,7}    {5,6,7,8}
                                             {4,6,7}
                                             {2,3,5,7}
                                             {2,5,6,7}
                                             {3,4,5,7}
                                             {3,5,6,7}
                                             {4,5,6,7}

First differences of A370584, cf. A370582, complement A370583.
For any number of choices we have A370586, complement A370587.
For binary indices see A370638, A370639, complement A370589.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
A370636 counts choosable subsets for binary indices, complement A370637.
Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905.

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

More terms from Jinyuan Wang, Mar 28 2025

A370590 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 5, 2, 4, 14, 25, 13, 38, 46, 66, 28, 178, 57, 235, 106, 238, 656, 1235, 288, 445, 2192, 664, 2016, 6840, 2300, 9140, 888, 6236, 17692, 14724, 7320, 56000, 60472, 70252, 37160, 223884, 66428, 290312, 113172, 80544, 517392, 1001420, 114336

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

For example, the set {4,7,9,10} has choice (2,7,3,5) so is counted under a(10).

			The a(0) = 0 through a(10) = 14 subsets (A = 10):
  .  .  2  23  34  235  256  2357  3578  2579  237A
                   345  356  2567  5678  4579  267A
                        456  3457        5679  279A
                             3567        5789  347A
                             4567              357A
                                               367A
                                               378A
                                               467A
                                               479A
                                               567A
                                               579A
                                               678A
                                               679A
                                               789A

Not requiring n gives A370585, maximal case of A370582, complement A370583.
Maximal case of A370586, complement A370587, unique A370588.
An opposite version is A370591.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A133686, A355739, A355744, A355745, A367771, A370584, A370636, A370639.

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

More terms from Jinyuan Wang, Feb 14 2025

A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 10, 13, 17, 21, 27, 34, 42, 53, 65, 80, 98, 119, 146, 177, 213, 258, 309, 370, 443, 528, 628, 745, 882, 1043, 1229, 1447, 1700, 1993, 2333, 2727, 3182, 3707, 4311, 5008, 5808, 6727, 7782, 8990, 10371, 11952, 13756, 15815, 18161

Offset: 0

Gus Wiseman, Sep 02 2025

First differs from A052337 in having 745 instead of 746.
We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct strict integer partitions of each part.
Also the number of integer partitions of n with no part k whose multiplicity exceeds A000009(k).

			The partition y = (3,3,2) has sets of strict integer partitions ({(2,1),(3)},{(2,1),(3)},{(2)}), and we have the choice ((2,1),(3),(2)) or ((3),(2,1),(2)), so y is counted under a(8).
The a(1) = 1 through a(9) = 10 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,3,1)  (4,4,1)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
                                                            (3,3,2,1)

For initial intervals instead of strict partitions we have A238873, ranks A387112.
For divisors instead of strict partitions we have A239312, ranks A368110.
The complement for divisors is A370320, ranks A355740.
For prime factors instead of strict partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement for initial intervals is A387118, ranks A387113.
The complement for all partitions is A387134, ranks A387577.
The complement is counted by A387137, ranks A387176.
These partitions are ranked by A387177.
For all partitions instead of just strict partitions we have A387328, ranks A387576.
The complement for constant partitions is A387329, ranks A387180.
For constant partitions instead of strict partitions we have A387330, ranks A387181.
A000041 counts integer partitions, strict A000009.
A358914 counts twice-partitions into distinct strict partitions.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A367867, A367901, A387115.

strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Select[Tuples[strptns/@#],UnsameQ@@#&]!={}&]],{n,0,15}]

cp's OEIS Frontend

A387137 Number of integer partitions of n whose parts do not have choosable sets of strict integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A370806 Number of non-strict condensed integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A370807 Number of integer partitions of n into parts > 1 such that it is not possible to choose a different prime factor of each part.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A370811 Numbers such that more than one set can be obtained by choosing a different divisor of each prime index.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A370815 Number of integer factorizations of n into unordered factors > 1, such that only one set can be obtained by choosing a different divisor of each factor.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A370588 Number of subsets of {1..n} containing n such that only one set can be obtained by choosing a different prime factor of each element.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A370590 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

Views

Author

Keywords

Comments

Examples

Crossrefs