A239312 -id:A239312 - cp's OEIS frontend

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

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&]

A371132 Number of integer partitions of n with fewer distinct parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 6, 10, 14, 21, 28, 40, 53, 73, 96, 130, 170, 223, 288, 375, 480, 616, 780, 990, 1245, 1567, 1954, 2440, 3024, 3745, 4610, 5674, 6947, 8499, 10349, 12591, 15258, 18468, 22277, 26841, 32238, 38673, 46262, 55278, 65881, 78423, 93136, 110477

Offset: 0

Gus Wiseman, Mar 17 2024

nonn

The Heinz numbers of these partitions are given by A371179.

			The partition (4,3,1,1) has 3 distinct parts {1,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(0) = 0 through a(9) = 14 partitions:
  .  .  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
                  (22)  (32)  (33)   (43)    (44)     (54)
                        (41)  (42)   (52)    (53)     (63)
                              (222)  (61)    (62)     (72)
                              (411)  (322)   (332)    (81)
                                     (4111)  (422)    (333)
                                             (431)    (432)
                                             (611)    (441)
                                             (2222)   (522)
                                             (41111)  (621)
                                                      (3222)
                                                      (4311)
                                                      (6111)
                                                      (411111)

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
The complement counting all parts on the LHS is A371172, ranks A371165.
Counting all parts on the LHS gives A371173, ranks A371168.
The complement is counted by A371178, ranks A371177.
These partitions are ranked by A371179.
The strict case is A371180, complement A371128.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A003963, A239312, A319055, A355740, A370802, A370803, A370808, A370813, A371130, A371171.

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

A371167 Positive integers with more divisors (A000005) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 34, 36, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 60, 62, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 82, 84, 85, 88, 90, 92, 93, 96, 98, 99, 100, 102, 104, 105, 108, 110

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.

			The prime indices of 814 are {1,5,12}, and there are 8 divisors (1,2,11,22,37,74,407,814) and 7 distinct divisors of prime indices (1,2,3,4,5,6,12), so 814 is in the sequence.
The prime indices of 1859 are {5,6,6}, and there are 6 divisors (1,11,13,143,169,1859) and 5 distinct divisors of prime indices (1,2,3,5,6), so 1859 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}

For prime factors on the LHS we have A370348, counted by A371171.
The RHS is A370820, for prime factors instead of divisors A303975.
For (equal to) instead of (greater than) we get A371165, counted by A371172.
For (less than) instead of (greater than) we get A371166.
Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
Other inequalities: A371168 (A371173), 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.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

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

A000005(a(n)) > A370820(a(n)).

A387112 Numbers with (strictly) choosable initial intervals of prime indices.

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, 50, 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

Offset: 1

Gus Wiseman, Aug 23 2025

First differs from A371088 in having a(86) = 121.
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 choosable.

			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 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 not in the sequence.

Partitions of this type are counted by A238873, complement A387118.
For partitions instead of initial intervals we have A276078, complement A276079.
For prime factors instead of initial intervals we have A368100, complement A355529.
For divisors instead of initial intervals we have A368110, complement A355740.
These are all the positions of nonzero terms in A387111, complement A387134.
The complement is A387113.
For strict partitions instead of initial intervals we have A387176, complement A387137.
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, A387110, A383706.

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

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}]

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172

Offset: 1

Gus Wiseman, Aug 27 2025

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 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.

The complement for all partitions appears to be A276078, counted by A052335.
For all partitions we appear to have A276079, counted by A387134.
For divisors instead of strict partitions we have A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873, see A387111.
For initial intervals instead of strict partitions we have A387113, counted by A387118.
These are the positions of 0 in A387115.
Partitions of this type are counted by A387137, complement A387178.
The complement is A387177.
The version for constant partitions is A387180, counted by A387329.
The complement for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.

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

A357865 Number of integer partitions of n whose run-sums are not weakly increasing.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 5, 10, 13, 22, 31, 45, 57, 85, 115, 155, 199, 267, 344, 452, 577, 744, 940, 1191, 1486, 1877, 2339, 2910, 3595, 4442, 5453, 6688, 8162, 9960, 12089, 14662, 17698, 21365, 25703, 30869, 36961, 44207, 52728, 62801, 74644, 88587, 104930, 124113

Offset: 0

Gus Wiseman, Oct 19 2022

nonn

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			The a(0) = 0 through a(8) = 13 partitions:
  .  .  .  (21)  (31)  (32)   (42)    (43)     (53)
                       (41)   (51)    (52)     (62)
                       (221)  (321)   (61)     (71)
                       (311)  (411)   (331)    (332)
                              (2211)  (421)    (431)
                                      (511)    (521)
                                      (2221)   (611)
                                      (3211)   (3221)
                                      (4111)   (3311)
                                      (22111)  (4211)
                                               (5111)
                                               (22211)
                                               (32111)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The complement is counted by A304406, ranked by A357861.
Number of rows in A354584 summing to n that are not weakly decreasing.
These partitions are ranked by A357850.
The opposite (not weakly decreasing) version is A357878, ranked by A357876.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A098859, A239312, A275870, A304405, A304428, A304430, A353832, A353864, A357875.

Table[Length[Select[IntegerPartitions[n],!LessEqual@@Total/@Split[#]&]],{n,0,30}]

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}]

cp's OEIS Frontend

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

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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.

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

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A371132 Number of integer partitions of n with fewer distinct parts than distinct divisors of parts.

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A371167 Positive integers with more divisors (A000005) than distinct divisors of prime indices (A370820).

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A387112 Numbers with (strictly) choosable initial intervals of prime indices.

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

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A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

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A357865 Number of integer partitions of n whose run-sums are not weakly increasing.

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