A365663 -id:A365663 - cp's OEIS frontend

A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.

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

Offset: 3

Gus Wiseman, Nov 18 2023

			Triangle begins:
  1
  0  1
  0  0  2
  1  1  1  2
  1  0  1  1  3
  1  1  1  1  2  3
  1  1  1  2  2  2  4
  2  2  3  2  3  2  3  4
  2  2  3  2  3  3  3  3  5
  3  2  4  3  4  4  5  3  4  5
  3  3  5  4  4  5  5  5  4  4  6
  4  3  6  5  6  5  7  5  7  4  5  6
  5  5  7  7  8  7  8  8  7  7  5  5  7
  6  5  9  8 10  7 10  9 10  7  9  5  6  7
  7  7 10 10 12 11 11 11 12 10  9  9  6  6  8
  9  7 13 11 15 12 13 13 15 13 13  9 11  6  7  8
Row n = 9 counts the following strict partitions:
  (6,2,1)  (5,3,1)  (4,3,2)  (5,3,1)  (6,2,1)  (6,2,1)  (8,1)
                             (4,3,2)  (4,3,2)  (5,3,1)  (7,2)
                                                        (6,3)
                                                        (5,4)
Row n = 13 counts the following strict partitions (A=10, B=11, C=12):
  A21   931   841   751   652   751   841   931   A21  A21  C1
  7321  7321  832   742   643   7321  742   832   832  931  B2
  6421  5431  7321  6421  6421  652   7321  7321  742  841  A3
              6421  5431  5431  6421  643   643   652  751  94
              5431              5431  5431  6421            85
                                                            76

Column n = k is A004526.
Column k = 3 is A025148.
For subsets instead of partitions we have A365541, non-binary A365381.
The non-binary version is A365661, non-strict A365543.
The non-binary complement is A365663, non-strict A046663.
Row sums are A366741, non-strict A366738.
The non-strict version is A367404.
Cf. A000041, A088809, A093971, A122768, A108917, A284640, A304792, A364272, A364911, A365658.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], k]&]], {n,3,10}, {k,3,n}]

A366753 Number of integer partitions of n without all different sums of two-element submultisets.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 9, 11, 22, 27, 48, 61, 98, 123, 188, 237, 345, 435, 611, 765, 1046, 1305, 1741, 2165, 2840, 3502, 4527, 5562, 7083, 8650, 10908, 13255, 16545, 20016, 24763, 29834, 36587, 43911, 53514, 63964, 77445, 92239, 111015, 131753

Offset: 0

Gus Wiseman, Nov 07 2023

nonn

			The two-element submultisets of y = {1,1,1,2,2,3} are {1,1}, {1,2}, {1,3}, {2,2}, {2,3}, with sums 2, 3, 4, 4, 5, which are not all different, so y is counted under a(10).
The a(8) = 1 through a(13) = 11 partitions:
  (3221)  (32211)  (4321)    (33221)    (4332)      (43321)
                   (32221)   (43211)    (5331)      (53221)
                   (322111)  (322211)   (5421)      (53311)
                             (3221111)  (43221)     (54211)
                                        (322221)    (332221)
                                        (332211)    (432211)
                                        (432111)    (3222211)
                                        (3222111)   (3322111)
                                        (32211111)  (4321111)
                                                    (32221111)
                                                    (322111111)

Semiprime divisors are counted by A086971, distinct sums A366739.
The non-binary complement is A108917, strict A275972, ranks A299702.
These partitions have ranks A366740.
The non-binary version is A366754, strict A316402, ranks A299729.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with a subset-sum k, complement A046663.
A365661 counts strict partitions with a subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
A367096 lists semiprime divisors, row sums A076290.
Cf. A002033, A001358, A006827, A008967, A122768, A126796, A365923, A365924, A367093, A367095.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Total/@Union[Subsets[#,{2}]]&]],{n,0,30}]

A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2, 2, 2, 5, 3, 2, 3, 1, 3, 2, 3, 6, 3, 3, 4, 3, 3, 4, 3, 3, 8, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12, 7, 6, 7, 7, 7, 4, 7, 7, 7, 6, 7, 15, 8, 7, 8, 8, 8, 8, 8

Offset: 1

Clark Kimberling, Feb 05 2014

			First 13 rows:
1
0 1
1 1 2
1 0 1 2
1 1 1 1 3
2 2 1 2 2 4
2 2 2 2 2 2 5
3 2 3 1 3 2 3 6
3 3 4 3 3 4 3 3 8
5 4 5 4 3 4 5 4 5 10
5 5 5 5 5 5 5 5 5 5 12
7 6 7 7 7 4 7 7 7 6 7 15
8 7 8 8 8 8 8 8 8 8 7 8 18
T(12,4) = 7 counts these partitions:  [8,4], [8,3,1], [7,4,1], [6,4,2], [6,3,2,1], [5,4,3], [5,4,2,1].

Clark Kimberling, Table of n, a(n) for n = 1..1000

Column k = n is A000009.
Column k = 2 is A015744.
Column k = 1 is A025147.
The non-strict complement is obtained by adding zeros after A046663.
Diagonal n = 2k is A237258.
Row sums are A284640.
For subsets instead of partitions we have A365381.
The non-strict version is obtained by removing column k = 0 from A365543.
Including column k = 0 gives A365661.
The complement is obtained by adding zeros after A365663.
Cf. A002219, A006827, A108796, A108917, A122768, A275972, A299701, A304792, A364272.

Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[nn], # == DeleteDuplicates[#] &]]; Table[Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], sumTo]] >= 1 &]]], {sumTo, nn}], {nn, 45}] // TableForm
u = Flatten[%]  (* Peter J. C. Moses, Feb 04 2014 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n,6}, {k,n}] (* Gus Wiseman, Nov 16 2023 *)

T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.

A365828 Number of strict integer partitions of 2n not containing n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55, 78, 108, 148, 201, 270, 359, 475, 623, 811, 1050, 1351, 1728, 2201, 2789, 3517, 4418, 5527, 6887, 8553, 10585, 13055, 16055, 19685, 24065, 29343, 35685, 43287, 52387, 63253, 76200, 91605, 109897, 131575, 157231, 187539

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

			The a(0) = 1 through a(6) = 12 strict partitions:
  ()  (2)  (4)    (6)    (8)      (10)       (12)
           (3,1)  (4,2)  (5,3)    (6,4)      (7,5)
                  (5,1)  (6,2)    (7,3)      (8,4)
                         (7,1)    (8,2)      (9,3)
                         (5,2,1)  (9,1)      (10,2)
                                  (6,3,1)    (11,1)
                                  (7,2,1)    (5,4,3)
                                  (4,3,2,1)  (7,3,2)
                                             (7,4,1)
                                             (8,3,1)
                                             (9,2,1)
                                             (5,4,2,1)

The complement is counted by A111133.
For non-strict partitions we have A182616, complement A000041.
A000009 counts strict integer partitions.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A365827 counts strict partitions not of length 2, complement A140106.
Cf. A008967, A035363, A078408, A079122, A231429, A238628, A344415, A365659.

Table[Length[Select[IntegerPartitions[2n],UnsameQ@@#&&FreeQ[#,n]&]],{n,0,30}]

a(n) = A000009(2n) - A000009(n) + 1.

A365919 Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.

Original entry on oeis.org

1, 3, 9, 21, 22, 27, 63, 76, 81, 117, 147, 175, 186, 189, 243, 248, 273, 286, 290, 322, 345, 351, 399, 418, 441, 513, 516, 567, 688, 715, 729, 819, 1029, 1053, 1062, 1156, 1180, 1197, 1323, 1375, 1416, 1484, 1521, 1539, 1701, 1827, 1888, 1911, 2068, 2115, 2130

Offset: 1

Gus Wiseman, Sep 25 2023

nonn

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

			The terms together with their prime indices begin:
     1: {}
     3: {2}
     9: {2,2}
    21: {2,4}
    22: {1,5}
    27: {2,2,2}
    63: {2,2,4}
    76: {1,1,8}
    81: {2,2,2,2}
   117: {2,2,6}
   147: {2,4,4}
   175: {3,3,4}
   186: {1,2,11}
   189: {2,2,2,4}
   243: {2,2,2,2,2}

The LHS is A304793, counted by A365658, with empty sets A299701.
The RHS is A325799, counted by A365923 (strict A365545).
A046663 counts partitions without a subset summing to k, strict A365663.
A056239 adds up prime indices, row sums of A112798.
A276024 counts positive subset-sums of partitions, strict A284640.
A325781 ranks complete partitions, counted by A126796.
A365830 ranks incomplete partitions, counted by A365924.
A365918 counts non-subset-sums of partitions, strict A365922.
Cf. A001223, A005117, A006827, A073491, A188431, A304792, A365831.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
smu[y_]:=Union[Total/@Rest[Subsets[y]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Select[Range[100],Length[smu[prix[#]]]==Length[nmz[prix[#]]]&]

Positive integers k such that A304793(k) = A325799(k).

A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of integer partitions of n with no two possibly equal parts summing to n.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (111)  (1111)  (221)    (222)     (322)      (332)
                           (311)    (411)     (331)      (521)
                           (2111)   (2211)    (421)      (611)
                           (11111)  (21111)   (511)      (2222)
                                    (111111)  (2221)     (3221)
                                              (3211)     (3311)
                                              (4111)     (5111)
                                              (22111)    (22211)
                                              (31111)    (32111)
                                              (211111)   (221111)
                                              (1111111)  (311111)
                                                         (2111111)
                                                         (11111111)

First condition alone is A058984, complement A004526, ranks A100959.
Second condition alone is A086543, complement A035363, ranks !A344415.
The complement is counted by A238628.
The strict case is A365826, complement A365659.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A140106 counts strict partitions of length 2, complement A365827.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A005408, A008967, A068911, A365377, A365544, A365543.

Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]

from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023

Heinz numbers are A100959 /\ !A344415.

a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - Chai Wah Wu, Sep 23 2023

a(31)-a(47) from Chai Wah Wu, Sep 23 2023

A365827 Number of strict integer partitions of n whose length is not 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 25, 30, 38, 45, 55, 66, 79, 93, 111, 130, 153, 179, 209, 242, 282, 325, 375, 432, 496, 568, 651, 742, 846, 963, 1094, 1240, 1406, 1589, 1795, 2026, 2282, 2567, 2887, 3240, 3634, 4072, 4557, 5094, 5692, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n with no pair of distinct parts summing to n.

			The a(5) = 1 through a(13) = 12 strict partitions (A..D = 10..13):
  (5)  (6)    (7)    (8)    (9)    (A)     (B)     (C)     (D)
       (321)  (421)  (431)  (432)  (532)   (542)   (543)   (643)
                     (521)  (531)  (541)   (632)   (642)   (652)
                            (621)  (631)   (641)   (651)   (742)
                                   (721)   (731)   (732)   (751)
                                   (4321)  (821)   (741)   (832)
                                           (5321)  (831)   (841)
                                                   (921)   (931)
                                                   (5421)  (A21)
                                                   (6321)  (5431)
                                                           (6421)
                                                           (7321)

The complement is counted by A140106 shifted left.
Heinz numbers are A005117 \ A006881 = A005117 /\ A100959.
The non-strict version is A058984, complement A004526.
The case not containing n/2 is A365826, non-strict A365825.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A008967, A035363, A078408, A365659.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[#]!=2&]],{n,0,30}]

a(n) = A000009(n) - A004526(n-1) for n > 0.

A367410 Number of strict integer partitions of n whose semi-sums cover an interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 6, 6, 7, 7, 8, 8, 11, 9, 11, 11, 12, 12, 15, 14, 15, 16, 16, 16, 19, 18, 19, 22, 21, 21, 24, 22, 25, 26, 26, 26, 30, 28, 29, 32, 31, 32, 37, 35, 36, 38, 39, 39, 43, 42, 43, 47, 46, 49, 51, 52, 51, 58

Offset: 0

Gus Wiseman, Nov 18 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is not counted under a(7).
The a(1) = 1 through a(9) = 6 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
                          (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
                                 (3,2,1)  (6,1)  (7,1)  (7,2)
                                                        (8,1)
                                                        (4,3,2)

For parts instead of sums we have A001227:
- non-strict A034296, ranks A073491
- complement A238007
- non-strict complement A239955, ranks A073492
The non-binary version is A188431:
- non-strict A126796, ranks A325781
- complement A365831
- non-strict complement A365924, ranks A365830
The non-strict version is A367402.
The non-strict complement is A367403.
The complement is counted by A367411.
A000009 counts partitions covering an initial interval, ranks A055932.
A046663 counts partitions w/o submultiset summing to k, strict A365663.
A365543 counts partitions w/ submultiset summing to k, strict A365661.
Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#,{2}]; If[d=={},{}, Range[Min@@d, Max@@d]]==Union[d])&]], {n,0,30}]

A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 4, 5, 8, 10, 14, 16, 23, 27, 35, 42, 52, 61, 75, 89, 106, 126, 149, 173, 204, 237, 274, 319, 369, 424, 490, 560, 642, 734, 838, 952, 1085, 1231, 1394, 1579, 1784, 2011, 2269, 2554, 2872, 3225, 3619, 4054, 4540, 5077, 5671, 6332

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is counted under a(7).
The a(7) = 1 through a(13) = 10 partitions:
  (4,2,1)  (4,3,1)  (5,3,1)  (5,3,2)  (5,4,2)  (6,4,2)    (6,4,3)
           (5,2,1)  (6,2,1)  (5,4,1)  (6,3,2)  (6,5,1)    (6,5,2)
                             (6,3,1)  (6,4,1)  (7,3,2)    (7,4,2)
                             (7,2,1)  (7,3,1)  (7,4,1)    (7,5,1)
                                      (8,2,1)  (8,3,1)    (8,3,2)
                                               (9,2,1)    (8,4,1)
                                               (5,4,2,1)  (9,3,1)
                                               (6,3,2,1)  (10,2,1)
                                                          (6,4,2,1)
                                                          (7,3,2,1)

For parts instead of sums we have A238007:
- complement A001227
- non-strict complement A034296, ranks A073491
- non-strict  A239955, ranks A073492
The non-strict version is A367403.
The non-strict complement is A367402.
The complement is counted by A367410.
The non-binary version is A365831:
- non-strict complement A126796, ranks A325781
- complement A188431
- non-strict A365924, ranks A365830
A000009 counts partitions covering an initial interval, ranks A055932.
A046663 counts partitions w/o submultiset summing to k, strict A365663.
A365543 counts partitions w/ submultiset summing to k, strict A365661.
Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#, {2}];If[d=={},{}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,30}]

A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 8, 4, 1, 0, 2, 19, 24, 6, 1, 0, 2, 47, 95, 49, 9, 1, 0, 6, 105, 363, 297, 93, 12, 1, 0, 12, 248, 1292, 1660, 753, 158, 16, 1, 0, 11, 563, 4649, 8409, 5591, 1653, 250, 20, 1, 0, 2, 1414, 15976, 41264, 38074, 15590, 3249, 380, 25, 1

Offset: 0

Gus Wiseman, Apr 16 2024

			The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
Triangle begins:
     1
     0     1
     0     1     1
     0     2     2     1
     0     2     8     4     1
     0     2    19    24     6     1
     0     2    47    95    49     9     1
     0     6   105   363   297    93    12     1
     0    12   248  1292  1660   753   158    16     1
     0    11   563  4649  8409  5591  1653   250    20     1
     0     2  1414 15976 41264 38074 15590  3249   380    25     1
Row n = 4 counts the following set partitions:
  .  {{1,4},{2,3}}  {{1},{2,3,4}}    {{1},{2},{3,4}}  {{1},{2},{3},{4}}
     {{1,2,3,4}}    {{1,2},{3},{4}}  {{1},{2,3},{4}}
                    {{1,2},{3,4}}    {{1},{2,4},{3}}
                    {{1,3},{2},{4}}  {{1,4},{2},{3}}
                    {{1,3},{2,4}}
                    {{1,2,3},{4}}
                    {{1,2,4},{3}}
                    {{1,3,4},{2}}

Row sums are A000110.
Column k = 1 is A035470.
A version for integer partitions is A116608.
For block lengths instead of sums we have A208437.
A008277 counts set partitions by length.
A275780 counts set partitions with distinct block-sums.
A371737 counts quanimous strict partitions, non-strict A321452.
A371789 counts non-quanimous sets, differences A371790.
Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]

cp's OEIS Frontend

A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.

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A366753 Number of integer partitions of n without all different sums of two-element submultisets.

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A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

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A365828 Number of strict integer partitions of 2n not containing n.

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A365919 Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.

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A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

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A365827 Number of strict integer partitions of n whose length is not 2.

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A367410 Number of strict integer partitions of n whose semi-sums cover an interval of positive integers.

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A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.

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