A324519 -id:A324519 - cp's OEIS frontend

A324522 Numbers > 1 where the minimum prime index is equal to the number of prime factors counted with multiplicity.

2, 9, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 125, 129, 141, 159, 175, 177, 183, 201, 213, 219, 237, 245, 249, 267, 275, 291, 303, 309, 321, 325, 327, 339, 381, 385, 393, 411, 417, 425, 447, 453, 455, 471, 475, 489, 501, 519, 537, 543, 573, 575, 579, 591

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of integer partitions where the minimum part is equal to the number of parts (A006141). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
    2: {1}
    9: {2,2}
   15: {2,3}
   21: {2,4}
   33: {2,5}
   39: {2,6}
   51: {2,7}
   57: {2,8}
   69: {2,9}
   87: {2,10}
   93: {2,11}
  111: {2,12}
  123: {2,13}
  125: {3,3,3}
  129: {2,14}
  141: {2,15}
  159: {2,16}
  175: {3,3,4}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001222, A006141, A055396, A056239, A106529, A112798, A256617.

Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(min(factorset(n)))=bigomega(n)):
select(q, [$2..600])[];  # Alois P. Heinz, Mar 07 2019

Select[Range[2,100],PrimePi[FactorInteger[#][[1,1]]]==PrimeOmega[#]&]

A055396(a(n)) = A001222(a(n)).

A324521 Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 112, 120, 125, 126, 128, 135, 140, 144, 150, 160, 162, 168, 176, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of integer partitions with nonnegative rank (A064174). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  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}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  40: {1,1,1,3}
  45: {2,2,3}
  48: {1,1,1,1,2}

Matthieu Pluntz, Table of n, a(n) for n = 1..10929 (up to a(n) = 2^21)

Cf. A001222, A003114, A056239, A061395, A064174, A106529, A112798, A256617.

Cf. A324515, A324517, A324519, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(max(factorset(n)))<=bigomega(n)):
select(q, [$2..300])[];  # Alois P. Heinz, Mar 07 2019

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]<=PrimeOmega[#]&]

isok(m) = (m>1) && (primepi(vecmax(factor(m)[, 1])) <= bigomega(m)); \\ Michel Marcus, Nov 14 2022

from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return primepi(max(f)) <= sum(f.values())
print([k for k in range(2, 257) if ok(k)]) # Michael S. Branicky, Nov 15 2022

A061395(a(n)) <= A001222(a(n)).

A324562 Numbers > 1 where the maximum prime index is greater than or equal to the number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of the integer partitions enumerated by A064174. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  28: {1,1,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A003114, A055396, A056239, A061395, A064174, A106529, A112798.

Cf. A324517, A324519, A324521, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(max(factorset(n)))>=bigomega(n)):
select(q, [$2..100])[];  # Alois P. Heinz, Mar 07 2019

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]>=PrimeOmega[#]&]

A061395(a(n)) >= A001222(a(n)).

A324517 Numbers > 1 where the maximum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

4, 24, 27, 36, 54, 80, 200, 224, 240, 360, 405, 500, 540, 600, 625, 672, 675, 704, 784, 810, 900, 1008, 1120, 1125, 1250, 1350, 1500, 1512, 1664, 1701, 1875, 2112, 2250, 2268, 2352, 2744, 2800, 3168, 3360, 3402, 3520, 3528, 3750, 3872, 3920, 3969, 4352, 4752

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of the integer partitions enumerated by A324518. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
   24: {1,1,1,2}
   27: {2,2,2}
   36: {1,1,2,2}
   54: {1,2,2,2}
   80: {1,1,1,1,3}
  200: {1,1,1,3,3}
  224: {1,1,1,1,1,4}
  240: {1,1,1,1,2,3}
  360: {1,1,1,2,2,3}
  405: {2,2,2,2,3}
  500: {1,1,3,3,3}
  540: {1,1,2,2,2,3}
  600: {1,1,1,2,3,3}
  625: {3,3,3,3}
  672: {1,1,1,1,1,2,4}
  675: {2,2,2,3,3}
  704: {1,1,1,1,1,1,5}
  784: {1,1,1,1,4,4}
  810: {1,2,2,2,2,3}
  900: {1,1,2,2,3,3}

Cf. A001221, A001222, A046660, A056239, A061395, A112798, A243055, A256617.

Cf. A324515, A324518, A324519, A324521, A324522, A324560, A324562.

Select[Range[2,1000],With[{f=FactorInteger[#]},PrimePi[f[[-1,1]]]==Total[Last/@f]-Length[f]]&]

A061395(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

A340692 Number of integer partitions of n of odd rank.

Original entry on oeis.org

0, 0, 2, 0, 4, 2, 8, 4, 14, 12, 26, 22, 44, 44, 76, 78, 126, 138, 206, 228, 330, 378, 524, 602, 814, 950, 1252, 1466, 1900, 2238, 2854, 3362, 4236, 5006, 6232, 7356, 9078, 10720, 13118, 15470, 18800, 22152, 26744, 31456, 37772, 44368, 53002, 62134, 73894

Offset: 0

Gus Wiseman, Jan 29 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The a(0) = 0 through a(9) = 12 partitions (empty columns indicated by dots):
  .  .  (2)   .  (4)     (32)   (6)       (52)     (8)         (54)
        (11)     (31)    (221)  (33)      (421)    (53)        (72)
                 (211)          (51)      (3211)   (71)        (432)
                 (1111)         (222)     (22111)  (422)       (441)
                                (411)              (431)       (621)
                                (3111)             (611)       (3222)
                                (21111)            (3221)      (3321)
                                (111111)           (3311)      (5211)
                                                   (5111)      (22221)
                                                   (22211)     (42111)
                                                   (41111)     (321111)
                                                   (311111)    (2211111)
                                                   (2111111)
                                                   (11111111)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of length/maximum instead of rank is A027193 (A026424/A244991).
The case of odd positive rank is A101707 is (A340604).
The strict case is A117193.
The even version is A340601 (A340602).
The Heinz numbers of these partitions are (A340603).
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064173 counts partitions of positive/negative rank (A340787/A340788).
A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).
A101198 counts partitions of rank 1 (A325233).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
- Odd -
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A003114, A006141, A027187, A039900, A067538, A096401, A117409, A143773, A324518, A325134, A340828, A340854/A340855.

Table[Length[Select[IntegerPartitions[n],OddQ[Max[#]-Length[#]]&]],{n,0,30}]

Having odd rank is preserved under conjugation, and self-conjugate partitions cannot have odd rank, so a(n) = 2*A101707(n) for n > 0.

A324520 Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.

Original entry on oeis.org

0, 1, 0, 1, 2, 2, 3, 3, 7, 6, 11, 12, 15, 21, 25, 31, 43, 49, 58, 79, 89, 108, 135, 165, 190, 232, 279, 328, 387, 461, 536, 650, 743, 870, 1029, 1202, 1381, 1613, 1864, 2163, 2505, 2875, 3292, 3829, 4367, 5001, 5746, 6538, 7462, 8533, 9714, 11008, 12527, 14196

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

The Heinz numbers of these integer partitions are given by A324519.

			The a(2) = 1 through a(11) = 11 integer partitions:
  (11)  (211)  (221)  (222)  (331)   (611)   (441)   (811)   (551)
               (311)  (411)  (511)   (3221)  (711)   (3322)  (911)
                             (3211)  (4211)  (3222)  (4222)  (3332)
                                             (3321)  (5221)  (4331)
                                             (4221)  (5311)  (4421)
                                             (4311)  (6211)  (5222)
                                             (5211)          (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)

Cf. A003114, A006141, A039900, A046660, A047993, A064174, A090858, A133121.

Cf. A324516, A324518, A324519, A324520.

Table[Length[Select[IntegerPartitions[n],Min@@#==Length[#]-Length[Union[#]]&]],{n,30}]

A340788 Heinz numbers of integer partitions of negative rank.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
      4: (1,1)             80: (3,1,1,1,1)
      8: (1,1,1)           81: (2,2,2,2)
     12: (2,1,1)           90: (3,2,2,1)
     16: (1,1,1,1)         96: (2,1,1,1,1,1)
     18: (2,2,1)          100: (3,3,1,1)
     24: (2,1,1,1)        108: (2,2,2,1,1)
     27: (2,2,2)          112: (4,1,1,1,1)
     32: (1,1,1,1,1)      120: (3,2,1,1,1)
     36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
     40: (3,1,1,1)        135: (3,2,2,2)
     48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
     54: (2,2,2,1)        150: (3,3,2,1)
     60: (3,2,1,1)        160: (3,1,1,1,1,1)
     64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
     72: (2,2,1,1,1)      168: (4,2,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 is (A340929).
The even case is A101708 is (A340930).
The positive version is (A340787).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]

For all terms A061395(a(n)) < A001222(a(n)).

A324560 Numbers > 1 where the minimum prime index is less than or equal to the number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of a certain type of integer partitions counted by A039900 (but not the type of partitions described in the name). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   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}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}

Cf. A001222, A039900, A055396, A056239, A061395, A106529, A112798.

Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

with(numtheory):
q:= n-> is(pi(min(factorset(n)))<=bigomega(n)):
select(q, [$2..100])[];  # Alois P. Heinz, Mar 07 2019

Select[Range[2,100],PrimePi[FactorInteger[#][[1,1]]]<=PrimeOmega[#]&]

A055396(a(n)) <= A001222(a(n)).

A340787 Heinz numbers of integer partitions of positive rank.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
     3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)
     5: (3)      29: (10)       51: (7,2)      70: (4,3,1)
     7: (4)      31: (11)       52: (6,1,1)    71: (20)
    10: (3,1)    33: (5,2)      53: (16)       73: (21)
    11: (5)      34: (7,1)      55: (5,3)      74: (12,1)
    13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)
    14: (4,1)    37: (12)       58: (10,1)     77: (5,4)
    15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)
    17: (7)      39: (6,2)      61: (18)       79: (22)
    19: (8)      41: (13)       62: (11,1)     82: (13,1)
    21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)
    22: (5,1)    43: (14)       65: (6,3)      85: (7,3)
    23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)
    25: (3,3)    46: (9,1)      67: (19)       87: (10,2)
    26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 (A340604).
The even case is A101708 (A340605).
The negative version is (A340788).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A200750 = partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]>PrimeOmega[#]&]

For all terms A061395(a(n)) > A001222(a(n)).

A324515 Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 40, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 97, 100, 101, 103, 107, 109, 112, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 175, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of the integer partitions enumerated by A324516. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  23: {9}
  29: {10}
  31: {11}
  37: {12}
  40: {1,1,1,3}
  41: {13}
  43: {14}
  45: {2,2,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006141, A046660, A047993, A055396, A056239, A061395, A106529, A112798, A243055.

Cf. A324516, A324517, A324519, A324521, A324522, A324560, A324562.

filter:= proc(n) local F, Inds, t;
  if isprime(n) then return true fi;
  F:= ifactors(n)[2];
  Inds:= map(numtheory:-pi, F[..,1]);
  max(Inds) - min(Inds) = add(t[2],t=F) - nops(F)
end proc:
select(filter, [$2..300]); # Robert Israel, Nov 19 2023

Select[Range[2,100],With[{f=FactorInteger[#]},PrimePi[f[[-1,1]]]-PrimePi[f[[1,1]]]==Total[Last/@f]-Length[f]]&]

A243055(a(n)) = A061395(a(n)) - A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

cp's OEIS Frontend

A324522 Numbers > 1 where the minimum prime index is equal to the number of prime factors counted with multiplicity.

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A324521 Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.

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A324562 Numbers > 1 where the maximum prime index is greater than or equal to the number of prime factors counted with multiplicity.

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A324517 Numbers > 1 where the maximum prime index equals the number of prime factors minus the number of distinct prime factors.

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A340692 Number of integer partitions of n of odd rank.

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A324520 Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.

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A340788 Heinz numbers of integer partitions of negative rank.

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