A243978 -id:A243978 - cp's OEIS frontend

A325242 Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 9, 10, 12, 11, 19, 15, 26, 1, 13, 39, 4, 25, 47, 5, 19, 70, 12, 29, 89, 17, 33, 115, 28, 42, 148, 41, 39, 189, 69, 62, 235, 88, 55, 294, 141, 81, 362, 183, 1, 84, 450, 253, 5, 103, 558, 333, 8, 105, 669, 464, 17, 153, 817, 576, 29

Offset: 1

Gus Wiseman, Apr 15 2019

For example, the partition (32111) has multiplicities {1,1,3}, of which 2 are distinct, so is counted under T(8,2).

			Triangle begins:
   1
   2
   3
   4   1
   4   3
   8   3
   6   9
  10  12
  11  19
  15  26   1
  13  39   4
  25  47   5
  19  70  12
  29  89  17
  33 115  28
  42 148  41
  39 189  69
  62 235  88
  55 294 141
  81 362 183   1
Row n = 8 counts the following partitions:
  (8)         (332)
  (44)        (422)
  (53)        (611)
  (62)        (3221)
  (71)        (4211)
  (431)       (5111)
  (521)       (22211)
  (2222)      (32111)
  (3311)      (41111)
  (11111111)  (221111)
              (311111)
              (2111111)

Alois P. Heinz, Rows n = 1..200, flattened

Row lengths are A056556. Row sums are A000041. Column k = 1 is A047966. Column k = 2 is A325243.

Cf. A008284, A062770, A071625, A098859, A116608, A127002, A183558, A243978, A244515, A325244, A325268.

DeleteCases[Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==k&]],{n,20},{k,n}],0,2]

A242451 Number T(n,k) of compositions of n in which the minimal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 1, 0, 1, 0, 15, 0, 0, 0, 1, 0, 23, 7, 1, 0, 0, 1, 0, 53, 10, 0, 0, 0, 0, 1, 0, 94, 32, 0, 1, 0, 0, 0, 1, 0, 203, 31, 21, 0, 0, 0, 0, 0, 1, 0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1, 0, 855, 77, 91, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1648, 222, 105, 71, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, May 15 2014

T(0,0) = 1 by convention. T(n,k) counts the compositions of n in which at least one part has multiplicity k and no part has a multiplicity smaller than k.
T(n,n) = T(2n,n) = 1.
T(3n,n) = A244174(n).

			T(5,1) = 15: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1], [2,3], [3,2], [1,4], [4,1], [5].
T(6,2) = 7: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3].
T(6,3) = 1: [2,2,2].
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,  1;
  0,   3,  0,  1;
  0,   6,  1,  0, 1;
  0,  15,  0,  0, 0, 1;
  0,  23,  7,  1, 0, 0, 1;
  0,  53, 10,  0, 0, 0, 0, 1;
  0,  94, 32,  0, 1, 0, 0, 0, 1;
  0, 203, 31, 21, 0, 0, 0, 0, 0, 1;
  0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A244164, A244165, A244166, A244167, A244168, A244169, A244170, A244171, A244172, A244173.
Row sums give A011782.
Cf. A242447 (the same for maximal multiplicity), A243978 (the same for partitions).

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
       b(n, i-1, p, k) +add(b(n-i*j, i-1, p+j, k)/j!,
       j=max(1, k)..floor(n/i))))
    end:
T:= (n, k)-> b(n$2, 0, k) -`if`(n=0 and k=0, 0, b(n$2, 0, k+1)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p, k] + Sum[b[n - i*j, i - 1, p + j, k]/j!, {j, Max[1, k], Floor[n/i]}]]]; T[n_, k_] := b[n, n, 0, k] - If[n == 0 && k == 0, 0, b[n, n, 0, k + 1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 1, 4, 3, 3, 2, 1, 0, 1, 3, 5, 4, 4, 3, 1, 1, 0, 1, 2, 6, 4, 8, 3, 3, 2, 1, 0, 1, 3, 7, 9, 6, 7, 4, 3, 1, 1, 0, 1, 1, 8, 7, 11, 9, 9, 4, 3, 2, 1

Offset: 0

Gus Wiseman, Nov 28 2023

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  1  1
  0  1  1  2  2  1
  0  1  3  3  2  1  1
  0  1  1  4  3  3  2  1
  0  1  3  5  4  4  3  1  1
  0  1  2  6  4  8  3  3  2  1
  0  1  3  7  9  6  7  4  3  1  1
  0  1  1  8  7 11  9  9  4  3  2  1
  0  1  5 10 11 13 10 11  6  5  3  1  1
  0  1  1 10 11 17 14 18 10  9  4  3  2  1
  0  1  3 12 17 19 18 22 14 12  8  4  3  1  1
  0  1  3 12 15 27 19 31 19 19 10  9  5  3  2  1
  0  1  4 15 23 27 31 33 24 26 18 12  8  4  3  1  1
  0  1  1 14 20 35 33 48 32 37 25 20 11 10  4  3  2  1
Row n = 7 counts the following partitions:
  (1111111)  (61)  (421)     (52)     (4111)  (511)  (7)
                   (2221)    (331)    (322)   (43)
                   (22111)   (31111)  (3211)
                   (211111)

Column k = 2 is A000005(n) - 1 = A032741(n).
Row sums are A000041.
The case of constant partitions is A051731, row sums A000005.
The corresponding rank statistic is A367581, row sums of A367579.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Cf. A000837, A056239, A066328, A071625, A072774, A082090, A367580.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n,0,10}, {k,0,n}]

A244515 Number of partitions of n where the minimal multiplicity of any part is 2.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 1, 4, 2, 6, 4, 9, 6, 16, 9, 23, 18, 34, 27, 51, 40, 75, 63, 103, 90, 152, 130, 208, 191, 286, 267, 402, 368, 546, 518, 730, 709, 998, 954, 1322, 1305, 1751, 1740, 2330, 2299, 3056, 3074, 3968, 4031, 5202, 5249, 6721, 6877, 8642, 8888, 11147, 11432, 14248, 14747, 18097, 18838, 23093, 23938, 29186, 30489

Offset: 1

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

			From _Gus Wiseman_, Jul 03 2019: (Start)
The a(2) = 1 through a(12) = 9 partitions are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325240.
  11  22  33    22111  44      33111    55        33311      66
          2211         3311    2211111  3322      44111      4422
                       22211            4411      3311111    5511
                       221111           222211    221111111  33222
                                        331111               332211
                                        22111111             441111
                                                             2222211
                                                             33111111
                                                             2211111111
(End)

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column k = 2 of A243978.

Cf. A000041, A007690, A008284, A116608, A325240, A325242.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))
    end:
a:= n-> b(n$2, 2) -b(n$2, 3):
seq(a(n), n=1..80);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1, k], {j, Max[1, k], n/i}]]];
a[n_] := b[n, n, 2] - b[n, n, 3];
Array[a, 80] (* Jean-François Alcover, May 01 2018, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[#]==2&]],{n,0,30}] (* Gus Wiseman, Jul 03 2019 *)

A367588 Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 5, 9, 6, 9, 10, 11, 8, 15, 9, 16, 14, 15, 11, 23, 14, 18, 18, 23, 14, 30, 15, 26, 22, 24, 22, 38, 18, 27, 26, 38, 20, 42, 21, 37, 36, 33, 23, 53, 27, 42, 34, 44, 26, 54, 34, 53, 38, 42, 29, 74, 30, 45, 49, 57, 40, 66, 33, 58, 46

Offset: 0

Gus Wiseman, Dec 01 2023

nonn

The Heinz numbers of these partitions are given by A268390.

			The a(3) = 1 through a(12) = 9 partitions (A = 10, B = 11):
  (21)  (31)  (32)  (42)    (43)  (53)    (54)      (64)    (65)  (75)
              (41)  (51)    (52)  (62)    (63)      (73)    (74)  (84)
                    (2211)  (61)  (71)    (72)      (82)    (83)  (93)
                                  (3311)  (81)      (91)    (92)  (A2)
                                          (222111)  (3322)  (A1)  (B1)
                                                    (4411)        (4422)
                                                                  (5511)
                                                                  (333111)
                                                                  (22221111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

For any multiplicities we have A002133, ranks A007774.
For any number of distinct parts we have A047966, ranks A072774.
For distinct multiplicities we have A182473, ranks A367589.
These partitions have ranks A268390.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
Cf. A023645, A091602, A116861, A181819, A243978, A367582.

Table[Sum[Floor[(d-1)/2],{d,Divisors[n]}],{n,30}]

G.f.: Sum_{i, j>0} x^(j*(2*i+1))/(1-x^j). - John Tyler Rascoe, Feb 04 2024

A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 3, 2, 5, 5, 10, 9, 14, 14, 21, 20, 30, 36, 44, 50, 66, 75, 93, 106, 132, 151, 185, 212, 256, 286, 348, 394, 479, 543, 642, 740, 888, 994, 1176, 1350, 1589, 1789, 2109, 2371, 2786, 3144, 3653, 4126, 4811, 5385, 6213

Offset: 0

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The a(1) = 1 through a(10) = 10 partitions:
  (1)  .  (21)  (22)   (41)   (51)    (61)   (71)     (81)    (91)
                (31)   (221)  (321)   (421)  (431)    (333)   (541)
                (211)         (3111)         (521)    (531)   (631)
                                             (3221)   (621)   (721)
                                             (41111)  (4221)  (3322)
                                                              (3331)
                                                              (4321)
                                                              (5221)
                                                              (322111)
                                                              (511111)

The case of strict partitions is A025147, ranks A039956.
The case of distinct multiplicities is A114640, ranks A109297.
These partitions have ranks A367683.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A000837, A071625, A072774, A082090, A116861, A367579, A367580, A367581.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]==mmk[#]&]], {n,0,15}]

A264397 Sum of the sizes of the longest clique of all partitions of n.

Original entry on oeis.org

1, 3, 5, 10, 15, 26, 38, 60, 86, 127, 178, 255, 349, 484, 652, 885, 1174, 1565, 2049, 2689, 3481, 4510, 5779, 7407, 9403, 11933, 15029, 18908, 23636, 29511, 36641, 45432, 56063, 69076, 84753, 103833, 126730, 154438, 187584, 227485, 275056, 332066, 399811

Offset: 1

Emeric Deutsch, Nov 20 2015

nonn

All parts of an integer partition with the same value form a clique. The size of a clique is the number of elements in the clique.

a(n) = Sum(k*A091602(n,k), k=1..n).

			a(4) = 10 because the partitions 4,31,22,211,1111 of 4 have longest clique sizes 1,1,2,2,4, respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A091602, A243978.

g := (sum(k*(product(1-x^(j*(k+1)), j = 1 .. 100) - product(1-x^(j*k), j = 1 .. 100)), k = 1 .. 100))/(product(1-x^j, j = 1 .. 100)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 1 .. 50);

from sympy.utilities.iterables import partitions
def A264397(n): return sum(max(p.values()) for p in partitions(n)) # Chai Wah Wu, Sep 17 2023

G.f.: g(x) = sum(k*(product(1-x^{j*(k+1)}, j>=1) - product(1-x^{j*k}, j>=1)), k>=1)/product(1-x^j, j>=1).

A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A177425 in lacking 360.
First differs from A182854 in lacking 360.
These are the Heinz numbers of the partitions counted by A182473.

			The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}

The case of any multiplicities is A007774, counts A002133.
These partitions are counted by A182473.
The case of equal exponents is A367590, counts A367588.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A098859 counts partitions with distinct multiplicities, ranks A130091.
A116608 counts partitions by number of distinct parts.
Cf. A071625, A072233, A072774, A109297, A367580.

Select[Range[100], PrimeNu[#]==2&&UnsameQ@@Last/@FactorInteger[#]&]

A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A268390 in lacking 210.
First differs from A238748 in lacking 210.
These are the Heinz numbers of the partitions counted by A367588.

			The terms together with their prime indices begin:
     6: {1,2}         57: {2,8}        106: {1,16}
    10: {1,3}         58: {1,10}       111: {2,12}
    14: {1,4}         62: {1,11}       115: {3,9}
    15: {2,3}         65: {3,6}        118: {1,17}
    21: {2,4}         69: {2,9}        119: {4,7}
    22: {1,5}         74: {1,12}       122: {1,18}
    26: {1,6}         77: {4,5}        123: {2,13}
    33: {2,5}         82: {1,13}       129: {2,14}
    34: {1,7}         85: {3,7}        133: {4,8}
    35: {3,4}         86: {1,14}       134: {1,19}
    36: {1,1,2,2}     87: {2,10}       141: {2,15}
    38: {1,8}         91: {4,6}        142: {1,20}
    39: {2,6}         93: {2,11}       143: {5,6}
    46: {1,9}         94: {1,15}       145: {3,10}
    51: {2,7}         95: {3,8}        146: {1,21}
    55: {3,5}        100: {1,1,3,3}    155: {3,11}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

The case of any multiplicities is A007774, counts A002133.
Partitions of this type are counted by A367588.
The case of distinct exponents is A367589, counts A182473.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A006881, A039956, A071625, A072233, A072774, A109297, A303661, A367580.

Select[Range[100], SameQ@@Last/@If[#==1, {}, FactorInteger[#]]&&PrimeNu[#]==2&]
Select[Range[200],PrimeNu[#]==2&&Length[Union[FactorInteger[#][[;;,2]]]]==1&] (* Harvey P. Dale, Aug 04 2025 *)

Union of A006881 and A303661. - Michael De Vlieger, Dec 01 2023

A367684 Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 14, 17, 25, 30, 39, 51, 66, 79, 102, 125, 154, 191, 233, 284, 347, 420, 499, 614, 726, 867, 1031, 1233, 1437, 1726, 2002, 2375, 2770, 3271, 3760, 4455, 5123, 5994, 6904, 8064, 9199, 10753, 12241, 14202, 16189, 18704, 21194, 24504

Offset: 0

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (2211)    (2221)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The case of strict partitions is A000012.
Includes all partitions with distinct multiplicities A098859, ranks A130091.
These partitions have ranks A367685.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Cf. A000837, A032741, A071625, A109297, A114640, A367579, A367580, A367581.

submultQ[cap_,fat_]:=And@@Function[i, Count[fat,i]>=Count[cap, i]]/@Union[List@@cap];
mmk[q_List]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], submultQ[mmk[#],#]&]], {n,0,15}]

cp's OEIS Frontend

A325242 Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

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A242451 Number T(n,k) of compositions of n in which the minimal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

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A244515 Number of partitions of n where the minimal multiplicity of any part is 2.

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A367588 Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.

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A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

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A264397 Sum of the sizes of the longest clique of all partitions of n.

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A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

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