A071364 -id:A071364 - cp's OEIS frontend

A071365 Numbers k such that A071364(k) <> A046523(k).

18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 198, 234, 242, 245, 250, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 414, 450, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 630, 648, 650, 666, 686, 702, 722, 726, 735, 738

Offset: 1

Reinhard Zumkeller, May 21 2002

nonn

A071364(k) and A046523(k) have the same prime factors, but not the same sequence of exponents in their prime factorizations.
A046523(k) <> k, as A046523(k) <= A071366(k) <= k.
Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is greater than the previous exponent when read from left to right; contains A097319. - Ray Chandler, Sep 23 2005
Choie et al. call the complementary set of integers (n = p1^e1 * p2^e^2 * ... with exponents e1 >= e2 >= e3 >= ... in their ordered prime factorization) Hardy-Ramanujan integers. - R. J. Mathar, Dec 08 2011
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 6, 71, 759, 7758, 77948, 780216, 7803437, 78033303, 780315757, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07803... . - Amiram Eldar, Aug 04 2024

			For k = 50 = 2*5*5: A071364(50) = 2*3*3 = 18, A046523(50) = 2*2*3 = 12.
For k = 500 = 2*2*5*5*5: A071364(500) = 2*2*3*3*3 = 108, A046523(500) = 2*2*2*3*3 = 72.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Y. Choie, N. Lichiardopol, P. Moree, and P. Sole, On Robin's criterion for the Riemann hypothesis, J. Theor. Nombr. Bord. 19 (2) (2007), 357-372

Cf. A046523, A071364, A071366.

a:= proc(n) option remember; local i, k, l;
      for k from 1 +`if`(n=1, 0, a(n-1))
      do l:= sort(ifactors(k)[2], (x, y)->x[1]Alois P. Heinz, Aug 18 2014

Select[Range[750], (e = Last /@ FactorInteger[ # ]) != Sort[e, Greater] &] (* Ray Chandler, Sep 23 2005 *)
Select[Range[750],
OrderedQ[FactorInteger[#][[All, 2]], GreaterEqual] == False &] (* Kenneth A Klinger, Nov 22 2016 *)

is(k) = {my(e = factor(k)[,2]); e != vecsort(e, , 4);} \\ Amiram Eldar, Aug 04 2024

Extended by Ray Chandler, Sep 23 2005

A083257 a(n) = A071364(n-th nonprime number) = A071364(A018252(n)).

Original entry on oeis.org

1, 4, 6, 8, 4, 6, 12, 6, 6, 16, 18, 12, 6, 6, 24, 4, 6, 8, 12, 30, 32, 6, 6, 6, 36, 6, 6, 24, 30, 12, 12, 6, 48, 4, 18, 6, 12, 54, 6, 24, 6, 6, 60, 6, 12, 64, 6, 30, 12, 6, 30, 72, 6, 18, 12, 6, 30, 48, 16, 6, 60, 6, 6, 6, 24, 90, 6, 12, 6, 6, 6, 96, 18, 12, 36, 30, 24, 30, 6, 108, 30, 6

Offset: 1

Labos Elemer, May 09 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007395, A000040, A002808, A018252, A046523, A071364, A083256.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Prime[Range[Length[e]]]^e)]; f[1] = 1; f /@ Select[Range[120], !PrimeQ[#] &] (* Amiram Eldar, Feb 22 2025 *)

Corrected by Ray Chandler, Aug 01 2004

A083260 a(n) = gcd(A046523(n), A071364(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 6, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 6, 6, 12, 2, 6, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 6, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2, 30, 6

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A071364, A083255, A083256, A083257, A083258, A083259, A083261.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A071364(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(i)); factorback(f); }; \\ This function from Michel Marcus, Jun 13 2014
A083260(n) = gcd(A046523(n), A071364(n)); \\ Antti Karttunen, Aug 24 2017

A083259 a(n) = gcd(n, A071364(n)), where A071364(n) is the smallest number with same sequence of exponents in canonical prime factorization as n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 2, 3, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 96, 1

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007395, A002808, A046523, A071364, A083255-A083260, A322361, A330749.

A071364(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = prime(i)); factorback(f); }; \\ From A071364
A083259(n) = gcd(A071364(n),n); \\ Antti Karttunen, Jan 22 2020

A071366 Numbers n such that A046523(n) <> A071364(n) and A071364(n) <> n.

Original entry on oeis.org

50, 75, 98, 126, 147, 198, 234, 242, 245, 250, 294, 306, 338, 342, 350, 363, 375, 378, 414, 490, 500, 507, 522, 525, 550, 558, 578, 588, 594, 605, 650, 666, 686, 702, 722, 726, 735, 738, 756, 774, 825, 845, 846, 847, 850, 867, 882, 918, 950, 954, 975, 980

Offset: 1

Reinhard Zumkeller, May 21 2002

nonn

Also A046523(n) <> n, as A046523(n) <= A071364(n) <= n.

			n=50=2*5*5: A071364(50)=2*3*3=18, A046523(50)=2*2*3=12;
n=500=2*2*5*5*5: A071364(500)=2*2*3*3*3=108, A046523(500)=2*2*2*3*3=72.

Cf. A071365.

Select[Range[1000], (e = Last /@ FactorInteger[ # ]) != Sort[e, Greater] && Product[Prime[i]^e[[i]], {i, Length[e]}] != # &] (* Ray Chandler, Sep 23 2005 *)

Extended by Ray Chandler, Sep 23 2005

A096011 Arrange numbers n such that A071364(n) <> A046523(n) by prime signature, cf. A071365.

Original entry on oeis.org

18, 54, 50, 90, 250, 75, 108, 126, 375, 98, 150, 500, 198, 686, 147, 162, 294, 1125, 234, 1029, 242, 270, 1250, 490, 1372, 306, 1715, 245, 300, 378, 1875, 726, 3087, 342, 2662, 338, 324, 588, 594, 4802, 735, 5324, 350, 3993, 363, 450, 2500, 980, 702

Offset: 0

Alford Arnold, Jul 20 2004

In the example, the first column is sequence A056808; the first row is A095990.

			The array begins
18 50 75 98 147 ...
54 250 375 686
90 126 198
108 500
150
...

Extended by Ray Chandler, Jul 31 2004

A357126 a(n) is the smallest positive integer k such that k > n and A071364(k) = A071364(n).

Original entry on oeis.org

3, 5, 9, 7, 10, 11, 27, 25, 14, 13, 20, 17, 15, 21, 81, 19, 50, 23, 28, 22, 26, 29, 40, 49, 33, 125, 44, 31, 42, 37, 243, 34, 35, 38, 100, 41, 39, 46, 56, 43, 66, 47, 45, 52, 51, 53, 80, 121, 75, 55, 63, 59, 250, 57, 88, 58, 62, 61, 84, 67, 65, 68, 729, 69, 70, 71, 76, 74, 78, 73, 200, 79, 77, 98

Offset: 2

Gleb Ivanov, Oct 26 2022

nonn

			a(12) = 20 as 12 has (2, 1) sequence of exponents in canonical prime factorization via 12 = 2^2 * 3^1 and the smallest positive integer > 12 with the same sequence of exponents in canonical prime factorization being (2, 1) is 20 as 20 = 2^2 * 5^1. - _David A. Corneth_, Oct 26 2022

Cf. A000961, A003961, A071364, A065642, A081382, A081761.

f4(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(i)); factorback(f); \\ A071364
a(n) = my(k=n+1, f=f4(n)); while (f4(k) != f, k++); k; \\ Michel Marcus, Oct 26 2022

first(n) = { my(res = vector(n + 1), todo = n, m = Map(), u = precprime(n)); for(e = 2, logint(n, 2), u = max(u, nextprime(sqrtnint(n, e) + 2)^e) ); forfactored(i = 2, u, cs = i[2][,2]; if(mapisdefined(m, cs), ci = mapget(m, cs); if(ci <= n + 1, res[ci] = i[1]; mapput(m, cs, i[1]); todo--; if(todo <= 0, res = res[^1]; return(res) ) ) , if(i[1] <= n + 1, mapput(m, cs, i[1]) ) ) ) } \\ David A. Corneth, Oct 26 2022

from sympy import factorint
to_s_exp = lambda n: tuple(i[1] for i in sorted(factorint(n).items()))
terms = []
for i in range(2, 100):
    k = i+1;t = to_s_exp(i)
    while t != to_s_exp(k):k+=1
    terms.append(k)
print(terms)

a(A000961(k)) = a(A003961(A000961(k))) for k > 1. - David A. Corneth, Oct 26 2022

a(n) >= A081761(n). - Rémy Sigrist, Feb 16 2023

A046523 Smallest number with same prime signature as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 12, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2

Offset: 1

N. J. A. Sloane

			If p,q,... are different primes, a(p)=2, a(p^2)=4, a(pq)=6, a(p^2*q)=12, etc.
n = 108 = 2*2*3*3*3 is replaced by a(n) = 2*2*2*3*3 = 72;
n = 105875 = 5*5*5*7*11*11 is represented by a(n) = 2*2*2*3*3*5 = 360.
Prime-powers are replaced by corresponding powers of 2, primes by 2.
Factorials, primorials and lcm[1..n] are in the sequence.
A000005(a(n)) = A000005(n) remains invariant; least and largest prime factors of a(n) are 2 or p[A001221(n)] resp.

T. D. Noe, Table of n, a(n) for n = 1..10000

A025487 gives range of values of this sequence.
Cf. A000142, A002110, A003418, A001221, A000040, A000005, A124010, A071364, A085079, A089247.
Cf. A181819, A181821.

import Data.List (sort)
a046523 = product .
          zipWith (^) a000040_list . reverse . sort . a124010_row
-- Reinhard Zumkeller, Apr 27 2013

a:= n-> (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)))
        (sort(ifactors(n)[2], (x, y)->x[2]>y[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 18 2014

Table[Apply[Times, p[w]^Reverse[Sort[ex[w]]]], {w, 1, 1000}] p[x_] := Table[Prime[w], {w, 1, lf[x]}] ex[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]]
ps[n_] := Sort[Last /@ FactorInteger[n]]; Join[{1}, Table[i = 2; While[ps[n] != ps[i], i++]; i, {n, 2, 89}]] (* Jayanta Basu, Jun 27 2013 *)

a(n)=my(f=vecsort(factor(n)[,2],,4),p);prod(i=1,#f,(p=nextprime(p+1))^f[i]) \\ Charles R Greathouse IV, Aug 17 2011

A046523(n)=factorback(primes(#n=vecsort(factor(n)[,2],,4)),n) \\ M. F. Hasler, Oct 12 2018, improved Jul 18 2019

from sympy import factorint
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1 # Indranil Ghosh, May 05 2017

from math import prod
from sympy import factorint, prime
def A046523(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n).values(),reverse=True))) # Chai Wah Wu, Feb 04 2022

In prime factorization of n, replace most common prime by 2, next most common by 3, etc.
a(n) = A124859(A124859(n)) = A181822(A124859(n)). - Matthew Vandermast, May 19 2012
a(n) = A181821(A181819(n)). - Alois P. Heinz, Feb 17 2020

Corrected and extended by Ray Chandler, Mar 11 2004

A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 6, 16, 2, 18, 2, 20, 10, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 40, 2, 42, 2, 44, 12, 46, 2, 48, 4, 50, 26, 52, 2, 54, 10, 56, 34, 58, 2, 60, 2, 62, 20, 64, 14, 66, 2, 68, 38, 70, 2, 72, 2

Offset: 1

Rémy Sigrist, Oct 31 2021

nonn

All terms except a(1) = 1 are even.
To compute a(n) for n > 1:
- if n = Product_{j = 1..o} prime(p_j)^e_j (where prime(i) denotes the i-th prime number, p_1 < ... < p_o and e_1 > 0)
- then a(n) = Product_{j = 1..o} prime(p_j + 1 - p_1)^e_j.
This sequence has similarities with A304776: here we shift down prime indexes, there prime exponents.
Smallest number generated by uniformly decrementing the indices of the prime factors of n. Thus, for n > 1, the smallest m > 1 such that the first differences of the indices of the ordered prime factors (including repetitions) are the same for m and n. As a function, a(.) preserves properties such as prime signature. - Peter Munn, May 12 2022

Positions of particular values (see formula section): A000040, A001248, A006094, A030078, A030514, A046301, A050997, A090076, A090090, A166329, A251720.
Also see formula section for the relationship to: A000265, A003961, A004277, A005940, A020639, A046523, A055396, A071364, A122111, A156552, A243055, A243074, A297845, A322993.
Sequences with comparable definitions: A304776, A316437.
Cf. A246277 (terms halved), A305897 (restricted growth sequence transform), A354185 (Möbius transform), A354186 (Dirichlet inverse), A354187 (sum with it).

a[1] = 1; a[n_] := Module[{f = FactorInteger[n], d}, d = PrimePi[f[[1, 1]]] - 1; Times @@ ((Prime[PrimePi[#[[1]]] - d]^#[[2]]) & /@ f)]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

a(n) = { my (f=factor(n)); if (#f~>0, my (pi1=primepi(f[1,1])); for (k=1, #f~, f[k,1] = prime(primepi(f[k,1])-pi1+1))); factorback(f) }

a(n) = n iff n belongs to A004277.
A003961^(A055396(n)-1)(a(n)) = n for any n > 1.
a(n) = 2 iff n belongs to A000040 (prime numbers).
a(n) = 4 iff n belongs to A001248 (squares of prime numbers).
a(n) = 6 iff n belongs to A006094 (products of two successive prime numbers).
a(n) = 8 iff n belongs to A030078 (cubes of prime numbers).
a(n) = 10 iff n belongs to A090076.
a(n) = 12 iff n belongs to A251720.
a(n) = 14 iff n belongs to A090090.
a(n) = 16 iff n belongs to A030514.
a(n) = 30 iff n belongs to A046301.
a(n) = 32 iff n belongs to A050997.
a(n) = 36 iff n belongs to A166329.
a(1) = 1, for n > 1, a(n) = 2*A246277(n). - Antti Karttunen, Feb 23 2022
a(n) = A122111(A243074(A122111(n))). - Peter Munn, Feb 23 2022
From Peter Munn and Antti Karttunen, May 12 2022: (Start)
a(1) = 1; a(2n) = 2n; a(A003961(n)) = a(n). [complete definition]
a(n) = A005940(1+A322993(n)) = A005940(1+A000265(A156552(n))).
Equivalently, A156552(a(n)) = A000265(A156552(n)).
A297845(a(n), A020639(n)) = n.
A046523(a(n)) = A046523(n).
A071364(a(n)) = A071364(n).
a(n) >= A071364(n).
A243055(a(n)) = A243055(n).
(End)

A057335 a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 60, 54, 90, 150, 210, 32, 48, 72, 120, 108, 180, 300, 420, 162, 270, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 240, 216, 360, 600, 840, 324, 540, 900, 1260, 1500, 2100, 2940, 4620, 486, 810, 1350, 1890, 2250, 3150, 4410

Offset: 0

Alford Arnold, Aug 27 2000

Note that for n>0 the prime divisors of a(n) are consecutive primes starting with 2. All of the least prime signatures (A025487) are included; with the other values forming A056808.
Using the formula, terms of b(n)= a(n)/A057334(n) are: 1, 1, 2, 2, 4, 4, 6, 6, 8, ..., indeed a(n) repeated. - Michel Marcus, Feb 09 2014
a(n) is the unique normal number whose unsorted prime signature is the k-th composition in standard order (graded reverse-lexicographic). This composition (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. A number is normal if its prime indices cover an initial interval of positive integers. Unsorted prime signature is the sequence of exponents in a number's prime factorization. - Gus Wiseman, Apr 19 2020

			From _Gus Wiseman_, Apr 19 2020: (Start)
The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     12: {1,1,2}
     18: {1,2,2}
     30: {1,2,3}
     16: {1,1,1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     60: {1,1,2,3}
     54: {1,2,2,2}
     90: {1,2,2,3}
    150: {1,2,3,3}
    210: {1,2,3,4}
     32: {1,1,1,1,1}
     48: {1,1,1,1,2}
For example, the 27th composition in standard order is (1,2,1,1), and the normal number with prime signature (1,2,1,1) is 630 = 2*3*3*5*7, so a(27) = 630.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000120, A057334, A055932 and A056808.
Cf. A324939.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
The reversed version is A334031.
A partial inverse is A334032.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A005867, A029931, A048793, A052409, A056239, A066099, A112798, A124767, A228351, A233249, A333220, A345974.
Related to A019565 via A122111 and to A000079 via A336321.

Table[Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ IntegerDigits[n, 2]], {n, 0, 54}] (* Michael De Vlieger, May 23 2017 *)

mg(n) = if (n==0, 1, prime(hammingweight(n))); \\ A057334
lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); v;} \\ Michel Marcus, Feb 09 2014

A057335(n) = if(0==n,1,prime(hammingweight(n))*A057335(n\2)); \\ Antti Karttunen, Jul 20 2020

a(n) = A057334(n) * a (repeated).
A334032(a(n)) = n; a(A334032(n)) = A071364(n). - Gus Wiseman, Apr 19 2020
a(n) = A122111(A019565(n)); A019565(n) = A122111(a(n)). - Peter Munn, Jul 18 2020
a(n) = A336321(2^n). - Peter Munn, Mar 04 2022
Sum_{n>=0} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

New primary name from Antti Karttunen, Jul 20 2020

cp's OEIS Frontend

A071365 Numbers k such that A071364(k) <> A046523(k).

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A083257 a(n) = A071364(n-th nonprime number) = A071364(A018252(n)).

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A083260 a(n) = gcd(A046523(n), A071364(n)).

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A083259 a(n) = gcd(n, A071364(n)), where A071364(n) is the smallest number with same sequence of exponents in canonical prime factorization as n.

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A071366 Numbers n such that A046523(n) <> A071364(n) and A071364(n) <> n.

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A096011 Arrange numbers n such that A071364(n) <> A046523(n) by prime signature, cf. A071365.

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A357126 a(n) is the smallest positive integer k such that k > n and A071364(k) = A071364(n).

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A046523 Smallest number with same prime signature as n.

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