A368110 -id:A368110 - cp's OEIS frontend

A370805 Number of condensed integer partitions of n into parts > 1.

1, 0, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 15, 18, 22, 27, 34, 41, 51, 62, 75, 90, 109, 129, 153, 185, 217, 258, 307, 359, 421, 493, 577, 675, 788, 909, 1062, 1227, 1418, 1633, 1894, 2169, 2497, 2860, 3285, 3754, 4298, 4894, 5587, 6359, 7230, 8215, 9331, 10567, 11965

Offset: 0

Gus Wiseman, Mar 04 2024

nonn

These are partitions without ones such that it is possible to choose a different divisor of each part.

			The a(0) = 1 through a(9) = 6 partitions:
  ()  .  (2)  (3)  (4)    (5)    (6)    (7)      (8)      (9)
                   (2,2)  (3,2)  (3,3)  (4,3)    (4,4)    (5,4)
                                 (4,2)  (5,2)    (5,3)    (6,3)
                                        (3,2,2)  (6,2)    (7,2)
                                                 (3,3,2)  (4,3,2)
                                                 (4,2,2)  (5,2,2)

The version with ones is A239312, complement A370320.
These partitions have as ranks the odd terms of A368110, complement A355740.
The version for prime factors is A370592, complement A370593, post A370807.
The complement without ones is A370804, ranked by the odd terms of A355740.
The version for factorizations is A370814, complement A370813.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355529, A355739, A367867, A367901, A368110, A368413, A370595, A370806, A370808, A370810.

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

More terms from Jinyuan Wang, Feb 14 2025

A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 62, 63, 64, 66, 68, 72, 75, 80, 81, 82, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 118, 120, 121, 124, 125, 126, 128, 132, 134, 135

Offset: 1

Gus Wiseman, Mar 16 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 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}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    22: {1,5}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    32: {1,1,1,1,1}
    34: {1,7}
    36: {1,1,2,2}

The strict version is A370348 counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The complement is A371168, counted by A371173.
The opposite version is A371170.
The version for prime factors instead of divisors on the RHS is A319899.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A355737, A355739, A355741, A370808, A370813, A370814, A371127.

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

A370816 Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 2, 4, 4, 7, 2, 7, 2, 7, 4, 4, 2, 11, 3, 4, 5, 7, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 11, 2, 8, 2, 7, 7, 4, 2, 17, 3, 7, 4, 7, 2, 11, 4, 11, 4, 4, 2, 15, 2, 4, 7, 14, 4, 8, 2, 7, 4, 8, 2, 20, 2, 4, 7, 7, 4, 8, 2, 17, 7, 4, 2

Offset: 1

Gus Wiseman, Mar 06 2024

nonn

			For the factorizations of 12 we have the following choices:
  (2*2*3): {{1,1,1},{1,1,2},{1,1,3},{1,2,2},{1,2,3},{2,2,3}}
    (2*6): {{1,1},{1,2},{1,3},{1,6},{2,2},{2,3},{2,6}}
    (3*4): {{1,1},{1,2},{1,3},{1,4},{2,3},{3,4}}
     (12): {{1},{2},{3},{4},{6},{12}}
So a(12) = 7.

Max Alekseyev, Table of n, a(n) for n = 1..10000

The version for partitions is A370808, for just prime factors A370809.
For just prime factors we have A370817.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A368413 counts non-choosable factorizations, complement A368414.
A370813 counts non-divisor-choosable factorizations, complement A370814.
Cf. A000792, A048249, A066739, A319055, A319057, A355737, A355739, A355740, A355741, A368110, A370803.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@facs[n]],{n,100}]

A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 74, 79, 89, 91, 95, 97, 101, 103, 106, 107, 111, 113, 122, 131, 137, 139, 141, 142, 143, 145, 149, 151, 159, 161, 163, 167, 169, 173, 178, 181, 183, 185, 193, 197, 199, 203, 209, 213, 214, 215, 219, 221, 223, 226

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 terms together with their prime indices begin:
     7: {4}       101: {26}      163: {38}      223: {48}
    13: {6}       103: {27}      167: {39}      226: {1,30}
    19: {8}       106: {1,16}    169: {6,6}     227: {49}
    23: {9}       107: {28}      173: {40}      229: {50}
    29: {10}      111: {2,12}    178: {1,24}    233: {51}
    37: {12}      113: {30}      181: {42}      239: {52}
    43: {14}      122: {1,18}    183: {2,18}    247: {6,8}
    47: {15}      131: {32}      185: {3,12}    251: {54}
    53: {16}      137: {33}      193: {44}      257: {55}
    61: {18}      139: {34}      197: {45}      259: {4,12}
    71: {20}      141: {2,15}    199: {46}      262: {1,32}
    73: {21}      142: {1,20}    203: {4,10}    263: {56}
    74: {1,12}    143: {5,6}     209: {5,8}     265: {3,16}
    79: {22}      145: {3,10}    213: {2,20}    267: {2,24}
    89: {24}      149: {35}      214: {1,28}    269: {57}
    91: {4,6}     151: {36}      215: {3,14}    271: {58}
    95: {3,8}     159: {2,16}    219: {2,21}    281: {60}
    97: {25}      161: {4,9}     221: {6,7}     293: {62}

The RHS is A370820, for prime factors instead of divisors A303975.
For (equal to) instead of (less than) we have A371165, counted by A371172.
For (greater than) instead of (less than) we have A371167.
For prime factors on the LHS we get A371168, counted by A371173.
Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
Other inequalities: A370348 (A371171), 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, A355737, A355739, A355741, A368100, A370808, A371127.

Select[Range[100],Length[Divisors[#]] < Length[Union@@Divisors/@PrimePi/@First/@FactorInteger[#]]&]

A000005(a(n)) < A370820(a(n)).

A387118 Number of integer partitions of n without choosable initial intervals.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 8, 13, 19, 28, 37, 52, 70, 97, 130, 172, 224, 293, 378, 492, 630, 806, 1018, 1286, 1609, 2019, 2514, 3131, 3874, 4784, 5872, 7198, 8786, 10712, 13013, 15794, 19100, 23063, 27752, 33341, 39939, 47781, 57013, 67955, 80816, 95992, 113773, 134668

Offset: 0

Gus Wiseman, Sep 02 2025

The initial interval of a nonnegative integer x is the set {1,...,x}.

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 partition y = (2,2,1) has initial intervals ({1,2},{1,2},{1}), which are not choosable, so y is counted under a(5).
The a(2) = 1 through a(8) = 13 partitions:
  (11)  (111)  (211)   (221)    (222)     (511)      (611)
               (1111)  (311)    (411)     (2221)     (2222)
                       (2111)   (2211)    (3211)     (3221)
                       (11111)  (3111)    (4111)     (3311)
                                (21111)   (22111)    (4211)
                                (111111)  (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement is counted by A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
For divisors instead of initial intervals we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of initial intervals we have A370593, ranks A355529.
These partitions have ranks A387113.
For partitions instead of initial intervals we have A387134.
The complement for partitions is A387328.
For strict partitions instead of initial intervals we have A387137, ranks A387176.
The complement for strict partitions is A387178.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
Cf. A005703, A052335, A367867, A367901, A367905, A367907, A370585, A370594, A387111.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Range/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

More terms from Jinyuan Wang, Sep 05 2025

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

Original entry on oeis.org

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&]

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)).

cp's OEIS Frontend

A370805 Number of condensed integer partitions of n into parts > 1.

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A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A370816 Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.

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A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

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A387118 Number of integer partitions of n without choosable initial intervals.

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

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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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