A325268 -id:A325268 - cp's OEIS frontend

A325280 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 3, 0, 0, 1, 3, 4, 3, 0, 0, 0, 1, 1, 4, 8, 1, 0, 0, 0, 1, 3, 6, 9, 3, 0, 0, 0, 0, 1, 2, 8, 12, 7, 0, 0, 0, 0, 0, 1, 3, 11, 17, 10, 0, 0, 0, 0, 0, 0, 1, 1, 11, 26, 17, 0, 0, 0, 0, 0, 0, 0, 1, 5, 19, 25, 27

Offset: 0

Gus Wiseman, Apr 18 2019

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (this sequence).

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  1  1
  0  1  1  2  3  0
  0  1  3  4  3  0  0
  0  1  1  4  8  1  0  0
  0  1  3  6  9  3  0  0  0
  0  1  2  8 12  7  0  0  0  0
  0  1  3 11 17 10  0  0  0  0  0
  0  1  1 11 26 17  0  0  0  0  0  0
  0  1  5 19 25 27  0  0  0  0  0  0  0
  0  1  1 17 44 38  0  0  0  0  0  0  0  0
  0  1  3 25 53 52  1  0  0  0  0  0  0  0  0
  0  1  3 29 63 76  4  0  0  0  0  0  0  0  0  0
  0  1  4 37 83 98  8  0  0  0  0  0  0  0  0  0  0
Row n = 9 counts the following partitions:
  (9)  (333)        (54)      (441)       (3321)
       (111111111)  (63)      (522)       (4221)
                    (72)      (711)       (4311)
                    (81)      (3222)      (5211)
                    (432)     (6111)      (32211)
                    (531)     (22221)     (42111)
                    (621)     (33111)     (321111)
                    (222111)  (51111)
                              (411111)
                              (2211111)
                              (3111111)
                              (21111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000041. Column k = 2 is A032741. Column k = 3 is A325245.
Cf. A181819, A225486, A323014, A323023, A325254, A325258, A325277.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or this sequence (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==k&]],{n,0,16},{k,0,n}]

\\ depth(p) gives adjusted frequency depth of partition.
depth(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L; r++); r)}
row(n)={my(v=vector(1+n)); forpart(p=n, v[1+depth(Vec(p))]++); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

A225485 Number of partitions of n that have frequency depth k, an array read by rows.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 4, 3, 1, 1, 4, 8, 1, 1, 3, 6, 9, 3, 1, 2, 8, 12, 7, 1, 3, 11, 17, 10, 1, 1, 11, 26, 17, 1, 5, 19, 25, 27, 1, 1, 17, 44, 38, 1, 3, 25, 53, 52, 1, 1, 3, 29, 63, 76, 4

Offset: 1

Clark Kimberling, May 08 2013

Let S = {x(1),...,x(k)} be a multiset whose distinct elements are y(1),...,y(h).  Let f(i) be the frequency of y(i) in S.  Define F(S) = {f(1),..,f(h)}, F(1,S) = F(S), and F(m,S) = F(F(m-1),S) for m>1.  Then lim(F(m,S)) = {1} for every S, so that there is a least positive integer i for which F(i,S) = {1}, which we call the frequency depth of S.
Equivalently, the frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). - Gus Wiseman, Apr 19 2019
From Clark Kimberling, Sep 26 2023: (Start)
Below, m^n abbreviates the sum m+...+m of n terms. In the following list, the numbers p_1,...,p_k are distinct, m >= 1, and k >= 1. The forms of the partitions being counted are as follows:
   column 1: [n],
   column 2: [m^k],
   column 3: [p_1^m,...,p_k^m],
   column 4: [(p_1^m_1)^m,..., (p_k^m_k)^m], distinct numbers m_i.
Column 3 is of special interest. Assume first that m = 1, so that the form of partition being counted is p = [p_1,...,p_k], with conjugate given by [q_1,...,q_m] where  q_i is the number of parts of p that are >= i. Since the p_i are distinct, the distinct parts of q are the integers 1,2,...,k. For the general case that m >= 1, the distinct parts of q are the integers m,...,km. Let S(n) denote the set of partitions of n counted by column 3. Then if a and b are in the set S*(n) of conjugates of partitions in S(n), and if a > b, then a - b is also in S*(n). Call this the subtraction property. Conversely, if a partition q has the subtraction property, then q must consist of a set of numbers m,..,km for some m. Thus, column 3 counts the partitions of n that have the subtraction property. (End)

			The first 9 rows:
  n = 1 .... 0
  n = 2 .... 1..1
  n = 3 .... 1..1..1
  n = 4 .... 1..2..1..1
  n = 5 .... 1..1..2..3
  n = 6 .... 1..3..4..3
  n = 7 .... 1..1..4..8..1
  n = 8 .... 1..3..6..9..3
  n = 9 .... 1..2..8.12..7
For the 7 partitions of 5, successive frequencies are shown here:
  5 -> 1 (depth 1)
  41 -> 11 -> 2 -> 1 (depth 3)
  32 -> 11 -> 2 -> 1 (depth 3)
  311 -> 12 -> 11 -> 2 -> 1 (depth 4)
  221 -> 12 -> 11 -> 2 -> 1 (depth 4)
  2111 -> 13 -> 11 -> 2 -> 1 (depth 4)
  11111 -> 5 -> 1 (depth 2)
Summary: 1 partition has depth 1; 1 has depth 2; 2 have 3; and 3 have 4, so that the row for n = 5 is 1..1..2..3 .

Alois P. Heinz, Rows n = 1..200, flattened (first 40 rows from Clark Kimberling)

Row sums are A000041.
Column k = 2 is A032741.
Column k = 3 is A325245.
a(n!) = A325272(n).
Cf. A181819, A182850, A225486, A323014, A323023, A325238, A325239, A325254.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
t[s_] := t[s] = Length[f[s]]
u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], i]],
  {i, 1, Length[IntegerPartitions[n]]}];
Flatten[Table[Count[u[n], k], {n, 2, 25}, {k, 1, Max[u[n]]}]]

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.

Original entry on oeis.org

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]

A325282 Maximum adjusted frequency depth among integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (A325280).
Run lengths are A325258, i.e., first differences of Levine's sequence A011784 (except at n = 1).

Cf. A011784, A032741, A127002, A181819, A225486, A275870, A323014, A323023, A325245, A325254, A325258, A325278, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Max@@fdadj/@IntegerPartitions[n],{n,0,30}]

a(0) = 0; a(1) = 1; a(n > 1) = A225486(n).

A325245 Number of integer partitions of n with adjusted frequency depth 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 11, 19, 17, 25, 29, 37, 37, 56, 53, 75, 80, 99, 103, 145, 143, 181, 199, 247, 255, 336, 339, 426, 459, 548, 590, 738, 759, 916, 999, 1192, 1259, 1529, 1609, 1915, 2083, 2406, 2589, 3085, 3267, 3809, 4134, 4763, 5119, 5964

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014.

			The a(3) = 1 through a(10) = 11 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)      (64)
              (41)  (51)    (52)   (62)    (63)      (73)
                    (321)   (61)   (71)    (72)      (82)
                    (2211)  (421)  (431)   (81)      (91)
                                   (521)   (432)     (532)
                                   (3311)  (531)     (541)
                                           (621)     (631)
                                           (222111)  (721)
                                                     (3322)
                                                     (4321)
                                                     (4411)

Column k = 3 of A225485 and A325280.

Cf. A008284, A047966, A116608, A127002, A181819, A182850, A323014, A323023, A325239, A325246, A325254, A325268, A325280.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==3&]],{n,0,30}]

A378972 Second differences of the strict partition numbers A000009.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 3, 2, 3, 4, 3, 4, 6, 4, 6, 8, 6, 9, 10, 9, 12, 14, 13, 16, 19, 18, 22, 26, 24, 30, 34, 34, 40, 45, 46, 53, 60, 62, 70, 79, 82, 93, 104, 108, 122, 136, 142, 160, 176, 186, 208, 228, 243, 268

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			The strict partition numbers begin (A000009):
  1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, ...
with differences (A087897 without first term):
  0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, ...
with differences (a(n)):
  0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, ...

For primes we have A036263.
The version for partitions is A053445.
For composites we have A073445.
For squarefree numbers we have A376590.
For nonsquarefree numbers we have A376593.
For powers of primes (inclusive) we have A376596.
For non powers of primes (inclusive) we have A376599.
Second row of A378622. See also:
- A293467 gives first column (up to sign).
- A377285 gives position of first zero in each row.
- A378970 gives row-sums.
- A378971 gives absolute value row-sums.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A225485, A325242, A325268.

Differences[Table[PartitionsQ[n],{n,0,100}],2]

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 3, 1, 3, 7, 10, 17, 27, 38, 1, 4, 8, 17, 31, 52, 83, 122, 181, 257, 361, 499, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 1, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754, 2570, 3742, 5269

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The Heinz numbers of these partitions are given by A325283.
The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014. The maximum adjusted frequency depth for integer partitions of n is given by A325282.
Essentially, the last numbers of rows of the array in A225485. - Clark Kimberling, Sep 13 2022

			The a(1) = 1 through a(11) = 17 partitions:
  1  11  21  211  221   411    3211  3221   3321    5221     4322
                  311   3111         4211   4221    5311     4331
                  2111  21111        32111  4311    6211     4421
                                            5211    32221    5411
                                            32211   33211    6221
                                            42111   42211    6311
                                            321111  43111    7211
                                                    52111    33221
                                                    421111   42221
                                                    3211111  43211
                                                             52211
                                                             53111
                                                             62111
                                                             431111
                                                             521111
                                                             4211111
                                                             32111111

Cf. A011784, A181819, A182850, A182857, A225486, A323014, A323023, A325246, A325258, A325278, A325281, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

nn=30;
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==mfds[[n]]&]],{n,0,nn}]

A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 3, 11, 2, 36, -27, 142, -207, 595, -1066, 2497, -4878, 10726, -22189, 48383, -103318, 224296, -480761, 1030299, -2186942, 4626313, -9740648, 20492711, -43109372, 90843475, -191769296, 405528200, -858373221, 1817311451, -3845483855, 8129033837

Offset: 0

Gus Wiseman, Dec 12 2024

sign

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 3.

For primes we have A140119 or A376683, unsigned A376681 or A376684.
These are the antidiagonal-sums of A175804.
First column of the same array is A281425.
For composites we have A377034, unsigned A377035.
For squarefree numbers we have A377039, unsigned A377040.
For nonsquarefree numbers we have A377049, unsigned A377048.
For prime powers we have A377052, unsigned A377053.
The unsigned version is A378621.
The version for strict partitions is A378970 (row-sums of A378622), unsigned A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A116608, A225486, A293467, A325242, A325245, A325268, A377285.

nn=20;
t=Table[Differences[PartitionsP/@Range[0,2nn],k],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

Original entry on oeis.org

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset: 0

Gus Wiseman, Dec 12 2024

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

For primes we have A376678.
For composites we have A377037.
For squarefree numbers we have A377042.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
Position of first zero in each row of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives first column (up to sign).
- A378970 gives row-sums.
- A378971 gives row-sums of absolute value.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]

a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, 8, 18, 30, 47, 70, 110, 177, 309, 574, 1063, 1892, 3107, 4598, 6166, 8737, 20603, 62457, 149132, 314116, 614093, 1155968, 2176048, 4244322, 8753864, 19006756, 42472117, 95235017, 210396059, 453414950, 949510166, 1931941261, 3826650257, 7400745917

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = 8.

For primes we have A376681 or A376684, signed version A140119 or A376683.
For composites we have A377035, signed version A377034.
For squarefree numbers we have A377040, signed version A377039.
For nonsquarefree numbers we have A377048, signed version A377049.
For prime powers we have A377053, signed version A377052.
For partition numbers we have A378621, signed version A377056.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The signed version is A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244, A325245, A325268.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

cp's OEIS Frontend

A325280 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.

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A225485 Number of partitions of n that have frequency depth k, an array read by rows.

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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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A325282 Maximum adjusted frequency depth among integer partitions of n.

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A325245 Number of integer partitions of n with adjusted frequency depth 3.

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A378972 Second differences of the strict partition numbers A000009.

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A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

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A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

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