cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 22, 24, 25, 27, 31, 34, 36, 40, 41, 44, 48, 50, 54, 59, 62, 67, 68, 72, 80, 81, 82, 83, 88, 96, 100, 108, 109, 118, 121, 124, 125, 127, 134, 136, 144, 157, 160, 162, 164, 166, 176, 179, 191, 192, 200, 211, 216, 218, 236, 241
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2024

Keywords

Comments

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.

Examples

			The terms together with their prime indices begin:
      3: {2}
      5: {3}
      6: {1,2}
      9: {2,2}
     10: {1,3}
     11: {5}
     12: {1,1,2}
     17: {7}
     18: {1,2,2}
     20: {1,1,3}
     22: {1,5}
     24: {1,1,1,2}
     25: {3,3}
     27: {2,2,2}
     31: {11}
     34: {1,7}
     36: {1,1,2,2}

Crossrefs

Subset of A302540.
Subset of A336101 = powers of 2 times powers of primes.
Positions of 2's in A370820.
Counting prime factors instead of divisors gives A371287.
A000005 counts divisors.
A000961 lists powers of primes, A302596 of prime index.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, indices A112798, length A001222.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000079, A005179, A007416, A303975, A319899, A355739, A370348, A370802, A371165-A371170, A371178.

Programs

  • Mathematica
    Select[Range[100],Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==2&]

A371178 Number of integer partitions of n containing all divisors of all parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 37, 48, 62, 80, 101, 127, 162, 202, 252, 312, 386, 475, 585, 713, 869, 1056, 1278, 1541, 1859, 2232, 2675, 3196, 3811, 4534, 5386, 6379, 7547, 8908, 10497, 12345, 14501, 16999, 19897, 23253, 27135, 31618, 36796, 42756
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371177.
Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.

Examples

			The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - _David A. Corneth_, Mar 18 2024
The a(0) = 1 through a(8) = 12 partitions:
  ()  (1)  (11)  (21)   (31)    (221)    (51)      (331)      (71)
                 (111)  (211)   (311)    (321)     (421)      (521)
                        (1111)  (2111)   (2211)    (511)      (3221)
                                (11111)  (3111)    (2221)     (3311)
                                         (21111)   (3211)     (4211)
                                         (111111)  (22111)    (5111)
                                                   (31111)    (22211)
                                                   (211111)   (32111)
                                                   (1111111)  (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)

Crossrefs

The LHS is represented by A001221, distinct case of A001222.
For partitions with no divisors of parts we have A305148, ranks A316476.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
Counting all parts on the LHS gives A371130, ranks A370802.
The complement is counted by A371132.
For submultisets instead of distinct parts we have A371172, ranks A371165.
These partitions have ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.

Programs

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

A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92
Offset: 1

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

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.

Examples

			The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
     1: {}       22: {1,5}      42: {1,2,4}    63: {2,2,4}
     2: {1}      23: {9}        43: {14}       65: {3,6}
     3: {2}      25: {3,3}      45: {2,2,3}    66: {1,2,5}
     5: {3}      26: {1,6}      46: {1,9}      67: {19}
     6: {1,2}    28: {1,1,4}    47: {15}       69: {2,9}
     7: {4}      29: {10}       49: {4,4}      70: {1,3,4}
     9: {2,2}    30: {1,2,3}    51: {2,7}      71: {20}
    10: {1,3}    31: {11}       52: {1,1,6}    73: {21}
    11: {5}      33: {2,5}      53: {16}       74: {1,12}
    13: {6}      34: {1,7}      55: {3,5}      75: {2,3,3}
    14: {1,4}    35: {3,4}      57: {2,8}      76: {1,1,8}
    15: {2,3}    37: {12}       58: {1,10}     77: {4,5}
    17: {7}      38: {1,8}      59: {17}       78: {1,2,6}
    19: {8}      39: {2,6}      61: {18}       79: {22}
    21: {2,4}    41: {13}       62: {1,11}     82: {1,13}

Crossrefs

The complement 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 strict version is A371168 counted by A371173.
The opposite version is A371169.
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, A319899, A355737, A355739, A355741, A370808, A371127.

Programs

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

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

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

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

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

Views

Author

Gus Wiseman, Mar 14 2024

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

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

Formula

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

A371131 Least number with exactly n distinct divisors of prime indices. Position of first appearance of n in A370820.

Original entry on oeis.org

1, 2, 3, 7, 13, 53, 37, 311, 89, 151, 223, 2045, 281, 3241, 1163, 827, 659, 9037, 1069, 17611, 1511, 4211, 28181, 122119, 2423, 10627, 88483, 6997, 7561, 98965, 5443, 88099, 6473, 95603, 309073, 50543, 10271, 192709, 508051, 438979, 14323, 305107, 26203
Offset: 0

Views

Author

Gus Wiseman, Mar 20 2024

Keywords

Comments

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.
Every nonnegative integer belongs to A370820, so this sequence is infinite.
Are there any terms with more than two prime factors?

Examples

			The terms together with their prime indices begin:
       1: {}
       2: {1}
       3: {2}
       7: {4}
      13: {6}
      53: {16}
      37: {12}
     311: {64}
      89: {24}
     151: {36}
     223: {48}
    2045: {3,80}
     281: {60}
    3241: {4,90}
    1163: {192}
     827: {144}
     659: {120}
    9037: {4,210}
    1069: {180}
   17611: {5,252}

Crossrefs

Counting prime factors instead of divisors (see A303975) gives A062447(>0).
The sorted version is A371181.
A000005 counts divisors.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000720, A000792, A005179, A007416, A355739, A370348, A370802, A370808, A371130, A371165, A371177.

Programs

  • Mathematica
    rnnm[q_]:=Max@@Select[Range[Min@@q,Max@@q],SubsetQ[q,Range[#]]&];
posfirsts[q_]:=Table[Position[q,n][[1,1]],{n,Min@@q,rnnm[q]}];
posfirsts[Table[Length[Union @@ Divisors/@PrimePi/@First/@If[n==1, {},FactorInteger[n]]],{n,1000}]]
  • PARI
    f(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i,1]), d, listput(list, d))); #Set(list); \\ A370820
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, May 02 2024

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

Views

Author

Gus Wiseman, Mar 17 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371179.

Examples

			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)

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Mar 14 2024

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

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

Formula

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

A371179 Positive integers with fewer distinct prime factors (A001221) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101
Offset: 1

Views

Author

Gus Wiseman, Mar 19 2024

Keywords

Comments

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.

Examples

			The terms together with their prime indices begin:
     3: {2}        28: {1,1,4}    52: {1,1,6}      74: {1,12}
     5: {3}        29: {10}       53: {16}         75: {2,3,3}
     7: {4}        31: {11}       55: {3,5}        76: {1,1,8}
     9: {2,2}      33: {2,5}      56: {1,1,1,4}    77: {4,5}
    11: {5}        35: {3,4}      57: {2,8}        78: {1,2,6}
    13: {6}        37: {12}       58: {1,10}       79: {22}
    14: {1,4}      38: {1,8}      59: {17}         81: {2,2,2,2}
    15: {2,3}      39: {2,6}      61: {18}         83: {23}
    17: {7}        41: {13}       63: {2,2,4}      85: {3,7}
    19: {8}        43: {14}       65: {3,6}        86: {1,14}
    21: {2,4}      45: {2,2,3}    67: {19}         87: {2,10}
    23: {9}        46: {1,9}      69: {2,9}        89: {24}
    25: {3,3}      47: {15}       70: {1,3,4}      91: {4,6}
    26: {1,6}      49: {4,4}      71: {20}         92: {1,1,9}
    27: {2,2,2}    51: {2,7}      73: {21}         93: {2,11}

Crossrefs

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
Partitions of this type are counted by A371132, strict A371180.
Counting all prime indices on the LHS gives A371168, counted by A371173.
The complement is A371177, counted by A371178, strict A371128.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts pairwise indivisible (stable) partitions, ranks A316476.
Cf. A003963, A239312, A285573, A355529, A370802, A370803, A371130, A371165, A371171, A371172.

Programs

  • Mathematica
    Select[Range[100],PrimeNu[#]

Formula

A001221(a(n)) < A370820(a(n)).

A371180 Number of strict integer partitions of n with fewer parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 2, 4, 4, 7, 8, 10, 12, 15, 19, 22, 29, 33, 40, 47, 57, 68, 81, 95, 110, 129, 152, 178, 207, 240, 277, 317, 365, 422, 486, 558, 632, 723, 824, 940, 1067, 1210, 1371, 1544, 1751, 1977, 2233, 2508, 2820, 3162, 3555, 3983, 4465, 4990, 5571, 6224
Offset: 0

Views

Author

Gus Wiseman, Mar 18 2024

Keywords

Examples

			The strict partition (6,4,2,1) has 4 parts and 5 distinct divisors of parts {1,2,3,4,5}, so is counted under a(13).
The a(2) = 1 through a(11) = 10 partitions:
  (2)  (3)  (4)  (5)    (6)    (7)    (8)      (9)      (10)     (11)
                 (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)
                 (4,1)         (5,2)  (6,2)    (6,3)    (7,3)    (7,4)
                               (6,1)  (4,3,1)  (7,2)    (8,2)    (8,3)
                                               (8,1)    (9,1)    (9,2)
                                               (4,3,2)  (5,3,2)  (10,1)
                                               (6,2,1)  (5,4,1)  (5,4,2)
                                                        (6,3,1)  (6,3,2)
                                                                 (6,4,1)
                                                                 (8,2,1)

Crossrefs

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
The version for equality is A371128.
The non-strict version is A371132, ranks A371179.
The non-strict complement is A371178, ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A003963, A239312, A319055, A355529, A370803, A370808, A370813, A371130 (A370802), A371171, A371172 (A371165), A371173 (A371168).

Programs

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