A370813 -id:A370813 - 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)).

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

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)).

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}]

A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 07 2024

nonn

First differs from A096825 at a(210) = 4, A096825(210) = 6.
First differs from A343943 at a(210) = 4, A343943(210) = 6.
First differs from A345926 at a(90) = 4, A345926(90) = 3.

			For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
  (2*2*3*5): {{1,1,2,3}}
   (2*2*15): {{1,1,2},{1,1,3}}
   (2*3*10): {{1,1,2},{1,2,3}}
    (2*5*6): {{1,1,3},{1,2,3}}
    (3*4*5): {{1,2,3}}
     (2*30): {{1,1},{1,2},{1,3}}
     (3*20): {{1,2},{2,3}}
     (4*15): {{1,2},{1,3}}
     (5*12): {{1,3},{2,3}}
     (6*10): {{1,1},{1,2},{1,3},{2,3}}
       (60): {{1},{2},{3}}
So a(60) = 4.

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

For all divisors (not just prime factors) we have A370816.
The version for partitions is A370809, for all divisors A370808.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
A368413 counts non-choosable factorizations, complement A368414.
A370813 counts non-divisor-choosable factorizations, complement A370814.
Cf. A000792, A048249, A066739, A319055, A319057, A355529, A355535, A367771, A368100, A370585.

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[If[#==1,{},First/@FactorInteger[#]]&/@#]]]&/@facs[n]],{n,100}]

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

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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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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.

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