A371177 -id:A371177 - cp's OEIS frontend

A370820 Number of positive integers that are a divisor of some prime index of n.

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

Offset: 1

Gus Wiseman, Mar 15 2024

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.

This sequence contains all nonnegative integers. In particular, a(prime(n)!) = n.

			2045 has prime indices {3,80} with combined divisors {1,2,3,4,5,8,10,16,20,40,80}, so a(2045) = 11. In fact, 2045 is the least number with this property.

a(prime(n)) = A000005(n).
Positions of ones are A000079 except for 1.
a(n!) = A000720(n).
a(prime(n)!) = a(prime(A005179(n))) = n.
Counting prime factors instead of divisors gives A303975.
Positions of 2's are A371127.
Position of first appearance of n is A371131(n), sorted version A371181.
RHS of A370802, A371128, A371130, A371165-A371170, A371177, A371178.
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. A000792, A007416, A048249, A319899, A355737, A355739, A370348, A370808, A370809.

Table[Length[Union@@Divisors/@PrimePi/@First/@If[n==1,{},FactorInteger[n]]],{n,100}]

a(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i,1]), d, listput(list, d))); #Set(list); \\ Michel Marcus, May 02 2024

A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 6, 9, 10, 22, 25, 28, 30, 34, 42, 45, 62, 63, 66, 75, 82, 92, 98, 99, 102, 104, 110, 118, 121, 134, 140, 147, 152, 153, 156, 166, 170, 186, 210, 218, 228, 230, 232, 234, 246, 254, 260, 275, 276, 279, 289, 308, 310, 314, 315, 330, 342, 343, 344, 348, 350

Offset: 1

Gus Wiseman, Mar 14 2024

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.

All squarefree terms are even.

			The prime indices of 1617 are {2,4,4,5}, with distinct divisors {1,2,4,5}, so 1617 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   42: {1,2,4}
   45: {2,2,3}
   62: {1,11}
   63: {2,2,4}
   66: {1,2,5}
   75: {2,3,3}
   82: {1,13}
   92: {1,1,9}
   98: {1,4,4}
   99: {2,2,5}
  102: {1,2,7}
  104: {1,1,1,6}

For factors instead of divisors on the RHS we have A319899.
A version for binary indices is A367917.
For (greater than) instead of (equal) we have A370348, counted by A371171.
The RHS is A370820, for prime factors instead of divisors A303975.
Partitions of this type are counted by A371130, strict A371128.
For divisors instead of factors on LHS we have A371165, counted by A371172.
For only distinct prime factors on LHS we have A371177, counted by A371178.
Other inequalities: A371166, A371167, 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.
Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.

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

A001222(a(n)) = A370820(a(n)).

A371130 Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 4, 2, 4, 5, 5, 11, 10, 16, 17, 21, 26, 32, 44, 53, 69, 71, 101, 110, 148, 168, 205, 249, 289, 356, 418, 502, 589, 716, 812, 999, 1137, 1365, 1566, 1873, 2158, 2537, 2942, 3449, 4001, 4613, 5380, 6193, 7220, 8224, 9575, 10926, 12683, 14430

Offset: 0

Gus Wiseman, Mar 17 2024

nonn

The Heinz numbers of these partitions are given by A370802.

			The partition (6,2,2,1) has 4 parts and 4 distinct divisors of parts {1,2,3,6} so is counted under a(11).
The a(1) = 1 through a(11) = 11 partitions:
  (1)  .  (21)  (22)  .  (33)   (322)  (71)   (441)   (55)    (533)
                (31)     (51)   (421)  (332)  (522)   (442)   (722)
                         (321)         (422)  (531)   (721)   (731)
                         (411)         (521)  (4311)  (4321)  (911)
                                              (6111)  (6211)  (4322)
                                                              (4331)
                                                              (5321)
                                                              (5411)
                                                              (6221)
                                                              (6311)
                                                              (8111)

The LHS is represented by A001222, distinct A000021.
These partitions are ranked by A370802.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
For (greater than) instead of (equal to) we have A371171, ranks A370348.
For submultisets instead of parts on the LHS we have A371172.
For (less than) instead of (equal to) we have A371173, ranked by A371168.
Counting only distinct parts on the LHS gives A371178, ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

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

A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336

Offset: 0

Gus Wiseman, Mar 18 2024

nonn

Also strict integer partitions such that the number of parts is equal to the number of distinct divisors of all parts.

			The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
  531  721   731   B1    751   D1    B31    D21    B51    H1     B71
       4321  5321  5421  931   B21   7521   7531   D31    9531   D51
                   6321  7321  7421  8421   64321  B321   A521   B521
                                     9321          65321  B421   D321
                                     54321         74321  75321  75421
                                                          84321  76321
                                                                 94321

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
Strict case of A371130 (ranks A370802) and A371178 (ranks A371177).
The complement is counted by A371180, non-strict A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A319055, A370803, A370808, A371171, A371172, A371173.

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

A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

3, 5, 11, 17, 26, 31, 35, 38, 39, 41, 49, 57, 58, 59, 65, 67, 69, 77, 83, 86, 87, 94, 109, 119, 127, 129, 133, 146, 148, 157, 158, 179, 191, 202, 206, 211, 217, 235, 237, 241, 244, 253, 274, 277, 278, 283, 284, 287, 291, 298, 303, 319, 326, 331, 333, 334, 353

Offset: 1

Gus Wiseman, Mar 14 2024

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.

			The terms together with their prime indices begin:
     3: {2}        67: {19}        158: {1,22}
     5: {3}        69: {2,9}       179: {41}
    11: {5}        77: {4,5}       191: {43}
    17: {7}        83: {23}        202: {1,26}
    26: {1,6}      86: {1,14}      206: {1,27}
    31: {11}       87: {2,10}      211: {47}
    35: {3,4}      94: {1,15}      217: {4,11}
    38: {1,8}     109: {29}        235: {3,15}
    39: {2,6}     119: {4,7}       237: {2,22}
    41: {13}      127: {31}        241: {53}
    49: {4,4}     129: {2,14}      244: {1,1,18}
    57: {2,8}     133: {4,8}       253: {5,9}
    58: {1,10}    146: {1,21}      274: {1,33}
    59: {17}      148: {1,1,12}    277: {59}
    65: {3,6}     157: {37}        278: {1,34}

For prime factors instead of divisors on both sides we get A319899.
For prime factors on LHS we get A370802, for distinct prime factors A371177.
The RHS is A370820, for prime factors instead of divisors A303975.
For (greater than) instead of (equal) we get A371166.
For (less than) instead of (equal) we get A371167.
Partitions of this type are counted by A371172.
Other inequalities: A370348 (A371171), 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.

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

A000005(a(n)) = A370820(a(n)).

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

Gus Wiseman, Mar 17 2024

nonn

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.

			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)

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.

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

A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 50, 54, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 108, 118, 124, 126, 128, 134, 136, 144, 160, 162, 164, 166, 168, 176, 192, 200, 216, 218, 230, 236, 242, 248, 250, 252, 254, 256, 268, 272, 288

Offset: 1

Gus Wiseman, Mar 22 2024

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.

			The prime indices of 694782 are {1,2,2,5,5,5,10} with distinct elements {1,2,5,10}, which form the set of divisors of 10, so 694782 is in the sequence.
The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   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}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Joseph Likar, Table of n, a(n) for n = 1..10000

The squarefree case is A371283, unsorted version A275700.
Partitions of this type are counted by A371284, strict A054973.
Products of squarefree terms are A371286, unsorted version A371285.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A003963, A005179, A007416, A034729, A370820, A371131, A371177.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Union[prix[#]]==Divisors[Max@@prix[#]]&]

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

Gus Wiseman, Mar 14 2024

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.

			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}

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.

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

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

A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.

Original entry on oeis.org

2, 6, 10, 22, 34, 42, 62, 82, 118, 134, 166, 218, 230, 254, 314, 358, 382, 390, 422, 482, 554, 566, 662, 706, 734, 798, 802, 862, 922, 1018, 1094, 1126, 1174, 1198, 1234, 1418, 1478, 1546, 1594, 1718, 1754, 1838, 1914, 1934, 1982, 2062, 2126, 2134, 2174, 2306

Offset: 1

Gus Wiseman, Mar 21 2024

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.

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.

			The terms together with their prime indices begin:
     2: {1}
     6: {1,2}
    10: {1,3}
    22: {1,5}
    34: {1,7}
    42: {1,2,4}
    62: {1,11}
    82: {1,13}
   118: {1,17}
   134: {1,19}
   166: {1,23}
   218: {1,29}
   230: {1,3,9}
   254: {1,31}
   314: {1,37}
   358: {1,41}
   382: {1,43}
   390: {1,2,3,6}

Joseph Likar, Table of n, a(n) for n = 1..10000

Partitions of this type are counted by A054973.
The unsorted version is A275700.
These numbers have products A371286, unsorted version A371285.
Squarefree case of A371288, counted by A371284.
A000005 counts divisors.
A001221 counts distinct prime factors.
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, A005179, A007416, A034729, A370820, A371131, A371177.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],SameQ[prix[#],Divisors[Last[prix[#]]]]&]

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

Gus Wiseman, Mar 20 2024

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

			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}

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.

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}]]

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

cp's OEIS Frontend

A370820 Number of positive integers that are a divisor of some prime index of n.

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A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A371130 Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.

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A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

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A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).

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

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A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

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A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

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A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.

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