A257541 -id:A257541 - cp's OEIS frontend

A340605 Heinz numbers of integer partitions of even positive rank.

5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
      5: (3)         57: (8,2)       97: (25)
     11: (5)         58: (10,1)      99: (5,2,2)
     14: (4,1)       59: (17)       102: (7,2,1)
     17: (7)         65: (6,3)      103: (27)
     21: (4,2)       66: (5,2,1)    104: (6,1,1,1)
     23: (9)         67: (19)       106: (16,1)
     26: (6,1)       68: (7,1,1)    109: (29)
     31: (11)        73: (21)       110: (5,3,1)
     35: (4,3)       74: (12,1)     111: (12,2)
     38: (8,1)       83: (23)       122: (18,1)
     39: (6,2)       86: (14,1)     124: (11,1,1)
     41: (13)        87: (10,2)     127: (31)
     44: (5,1,1)     91: (6,4)      129: (14,2)
     47: (15)        92: (9,1,1)    133: (8,4)
     49: (4,4)       95: (8,3)      137: (33)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
Allowing any positive rank gives A064173 (A340787).
The odd version is counted by A101707 (A340604).
These partitions are counted by A101708.
The not necessarily positive case is counted by A340601 (A340602).
A001222 counts prime indices.
A061395 gives maximum prime index.
A072233 counts partitions by sum and length.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A064173 counts partitions of negative rank (A340788).
A064174 counts partitions of nonnegative rank (A324562).
A064174 (also) counts partitions of nonpositive rank (A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340692 counts partitions of odd rank (A340603).
- Even -
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A035363 counts partitions into even parts (A066207).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A339846 counts factorizations of even length.
Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[100],EvenQ[rk[#]]&&rk[#]>0&]

A061395(a(n)) - A001222(a(n)) is even and positive.

A340604 \/ A340605 = A340787.

A340603 Heinz numbers of integer partitions of odd rank.

Original entry on oeis.org

3, 4, 7, 10, 12, 13, 15, 16, 18, 19, 22, 25, 27, 28, 29, 33, 34, 37, 40, 42, 43, 46, 48, 51, 52, 53, 55, 60, 61, 62, 63, 64, 69, 70, 71, 72, 76, 77, 78, 79, 82, 85, 88, 89, 90, 93, 94, 98, 100, 101, 105, 107, 108, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
      3: (2)           33: (5,2)           63: (4,2,2)
      4: (1,1)         34: (7,1)           64: (1,1,1,1,1,1)
      7: (4)           37: (12)            69: (9,2)
     10: (3,1)         40: (3,1,1,1)       70: (4,3,1)
     12: (2,1,1)       42: (4,2,1)         71: (20)
     13: (6)           43: (14)            72: (2,2,1,1,1)
     15: (3,2)         46: (9,1)           76: (8,1,1)
     16: (1,1,1,1)     48: (2,1,1,1,1)     77: (5,4)
     18: (2,2,1)       51: (7,2)           78: (6,2,1)
     19: (8)           52: (6,1,1)         79: (22)
     22: (5,1)         53: (16)            82: (13,1)
     25: (3,3)         55: (5,3)           85: (7,3)
     27: (2,2,2)       60: (3,2,1,1)       88: (5,1,1,1)
     28: (4,1,1)       61: (18)            89: (24)
     29: (10)          62: (11,1)          90: (3,2,2,1)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
These partitions are counted by A340692.
The complement is A340602, counted by A340601.
The case of positive rank is A340604.
- Rank -
A001222 gives number of prime indices.
A047993 counts partitions of rank 0 (A106529).
A061395 gives maximum prime index.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length (A026424).
A027193 (also) counts partitions of odd maximum (A244991).
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.
A340102 counts odd-length factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340608, A340609, A340610.

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

A061395(a(n)) - A001222(a(n)) is odd.

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)).

A325232 Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.

Original entry on oeis.org

2, 3, 6, 10, 18, 27, 44, 64, 97, 138, 200, 276, 390, 528, 724, 968, 1301, 1712, 2266, 2946, 3842, 4947, 6372, 8122, 10362, 13094, 16544, 20754, 26010, 32392, 40308, 49876, 61648, 75845, 93178, 114006, 139308, 169586, 206158, 249814, 302267, 364664, 439330

Offset: 0

Gus Wiseman, Apr 13 2019

nonn

			The a(0) = 1 through a(4) = 18 partitions:
  ()   (2)   (3)    (4)     (5)
  (1)  (11)  (22)   (32)    (33)
       (21)  (31)   (41)    (42)
             (111)  (221)   (51)
             (211)  (321)   (222)
             (311)  (411)   (322)
                    (1111)  (331)
                    (2111)  (421)
                    (3111)  (511)
                    (4111)  (2211)
                            (3211)
                            (4211)
                            (5111)
                            (11111)
                            (21111)
                            (31111)
                            (41111)
                            (51111)

Giovanni Resta, Table of n, a(n) for n = 0..75
FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Number of times n appears in A325224.

Cf. A051924, A257541, A263297, A325193, A325194, A325224, A325225, A325227.

nn=30;
mindif[ptn_]:=If[ptn=={},0,Total[ptn]-Min[Length[ptn],Max[ptn]]];
allip=Array[IntegerPartitions,2*nn+2,0,Join];
Table[Length[Select[allip,mindif[#]==n&]],{n,0,nn}]

For n > 0, a(n) = Sum_{k > 0} A325227(n + k, k).

More terms from Giovanni Resta, Apr 15 2019

A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 47, 53, 57, 58, 59, 61, 65, 67, 71, 73, 74, 78, 79, 83, 86, 87, 89, 91, 95, 97, 101, 103, 106, 107, 109, 111, 113, 122, 127, 129, 130, 131, 133, 137, 138, 139, 142, 143, 145

Offset: 1

Gus Wiseman, Feb 05 2021

nonn

Also Heinz numbers of strict integer partitions whose greatest part is divisible by their number of parts. These partitions are counted by A340828.

			The sequence of terms together with their prime indices begins:
      2: {1}         31: {11}       71: {20}
      3: {2}         35: {3,4}      73: {21}
      5: {3}         37: {12}       74: {1,12}
      6: {1,2}       38: {1,8}      78: {1,2,6}
      7: {4}         39: {2,6}      79: {22}
     11: {5}         41: {13}       83: {23}
     13: {6}         43: {14}       86: {1,14}
     14: {1,4}       47: {15}       87: {2,10}
     17: {7}         53: {16}       89: {24}
     19: {8}         57: {2,8}      91: {4,6}
     21: {2,4}       58: {1,10}     95: {3,8}
     23: {9}         59: {17}       97: {25}
     26: {1,6}       61: {18}      101: {26}
     29: {10}        65: {3,6}     103: {27}
     30: {1,2,3}     67: {19}      106: {1,16}

Note: Heinz number sequences are given in parentheses below.
The case of equality, and the reciprocal version, are both A002110.
The non-strict reciprocal version is A168659 (A340609).
The non-strict version is A168659 (A340610).
These are the Heinz numbers of partitions counted by A340828.
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up the prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413/A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A257541 gives the rank of the partition with Heinz number n.
A340830 counts strict partitions whose parts are multiples of the length.
Cf. A039900, A047993 (A106529), A064174, A143773 (A316428), A326837, A326849 (A326848), A340599, A340691.

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

A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 18, 21, 22, 24, 25, 26, 27, 33, 34, 35, 36, 38, 39, 46, 48, 49, 51, 54, 55, 57, 58, 62, 65, 69, 72, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 108, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 144, 145, 146

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

The enumeration of these partitions by sum is given by A265283.

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:
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   46: {1,9}

Cf. A056239, A061395, A093641, A112798, A252464, A257541, A263297, A265283, A325224, A325225, A325227, A325230, A325232.

Select[Range[300],Min[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==2&]

A325230 Numbers of the form p^k * q, p and q prime, p > q, k > 0.

Original entry on oeis.org

6, 10, 14, 15, 18, 21, 22, 26, 33, 34, 35, 38, 39, 46, 50, 51, 54, 55, 57, 58, 62, 65, 69, 74, 75, 77, 82, 85, 86, 87, 91, 93, 94, 95, 98, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 147, 155, 158, 159, 161, 162, 166, 177, 178

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   54: {1,2,2,2}
   55: {3,5}
   57: {2,8}
   58: {1,10}

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

Positions of 1's in A325226.

Cf. A056239, A093641, A112798, A174090, A257541, A307517, A325223, A325224, A325225, A325231.

filter:= proc(n) local F;
   F:= sort(ifactors(n)[2],(a,b)-> a[1]Robert Israel, Apr 14 2019

Select[Range[100],PrimeOmega[#/Power@@FactorInteger[#][[-1]]]==1&]

from sympy import factorint
A325230_list = [n for n, m in ((n, factorint(n)) for n in range(2,10**6)) if len(m) == 2 and m[min(m)] == 1] # Chai Wah Wu, Apr 16 2019

A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

Original entry on oeis.org

6, 10, 12, 14, 22, 24, 26, 34, 38, 46, 48, 58, 62, 74, 82, 86, 94, 96, 106, 118, 122, 134, 142, 146, 158, 166, 178, 192, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 384, 386, 394, 398, 422, 446, 454, 458, 466

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Also numbers n such that the sum of prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n is 1. 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, and their sum is A056239.

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   48: {1,1,1,1,2}
   58: {1,10}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
   96: {1,1,1,1,1,2}
  106: {1,16}
  118: {1,17}

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Positions of 1's in A325223.

Cf. A001222, A056239, A060687, A061395, A093641, A112798, A174090, A257541, A265283, A325224, A325225, A325227, A325232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[primeMS[#]]-Max[Length[primeMS[#]],Max[primeMS[#]]]==1&]

from sympy import isprime
A325231_list = [n for n in range(6,10**6) if ((not n % 2) and isprime(n//2)) or (bin(n)[2:4] == '11' and bin(n).count('1') == 2)] # Chai Wah Wu, Apr 16 2019

A325234 Heinz numbers of integer partitions with Dyson rank -1.

Original entry on oeis.org

4, 12, 18, 27, 40, 60, 90, 100, 112, 135, 150, 168, 225, 250, 252, 280, 352, 375, 378, 392, 420, 528, 567, 588, 625, 630, 700, 792, 832, 880, 882, 945, 980, 1050, 1188, 1232, 1248, 1320, 1323, 1372, 1470, 1575, 1750, 1782, 1848, 1872, 1936, 1980, 2058, 2080

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Numbers whose maximum prime index is one fewer than their number of prime indices counted with multiplicity.

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}
    12: {1,1,2}
    18: {1,2,2}
    27: {2,2,2}
    40: {1,1,1,3}
    60: {1,1,2,3}
    90: {1,2,2,3}
   100: {1,1,3,3}
   112: {1,1,1,1,4}
   135: {2,2,2,3}
   150: {1,2,3,3}
   168: {1,1,1,2,4}
   225: {2,2,3,3}
   250: {1,3,3,3}
   252: {1,1,2,2,4}
   280: {1,1,1,3,4}
   352: {1,1,1,1,1,5}
   375: {2,3,3,3}
   378: {1,2,2,2,4}
   392: {1,1,1,4,4}

FindStat, St000145: The Dyson rank of a partition

Positions of -1's in A257541.

Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325233, A325235.

Select[Range[1000],PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]==-1&]

A325235 Heinz numbers of integer partitions with Dyson rank 1 or -1.

Original entry on oeis.org

3, 4, 10, 12, 15, 18, 25, 27, 28, 40, 42, 60, 63, 70, 88, 90, 98, 100, 105, 112, 132, 135, 147, 150, 168, 175, 198, 208, 220, 225, 245, 250, 252, 280, 297, 308, 312, 330, 343, 352, 375, 378, 392, 420, 462, 468, 484, 495, 520, 528, 544, 550, 567, 588, 625, 630

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Numbers whose maximum prime index and number of prime indices counted with multiplicity differ by 1.

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:
    3: {2}
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   88: {1,1,1,5}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  105: {2,3,4}
  112: {1,1,1,1,4}

FindStat, St000145: The Dyson rank of a partition

Positions of 1's and -1's in A257541.

Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325233, A325234.

Select[Range[1000],Abs[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]==1&]

cp's OEIS Frontend

A340605 Heinz numbers of integer partitions of even positive rank.

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A340603 Heinz numbers of integer partitions of odd rank.

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A340787 Heinz numbers of integer partitions of positive rank.

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A325232 Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.

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A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

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A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

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A325230 Numbers of the form p^k * q, p and q prime, p > q, k > 0.

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A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

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