A105027 -id:A105027 - cp's OEIS frontend

A214414 Fixed points of permutations A105027 and A214417.

0, 1, 5, 7, 18, 22, 26, 30, 1031, 1039, 1047, 1055, 1063, 1071, 1079, 1087, 1095, 1103, 1111, 1119, 1127, 1135, 1143, 1151, 1159, 1167, 1175, 1183, 1191, 1199, 1207, 1215, 1223, 1231, 1239, 1247, 1255, 1263, 1271, 1279, 1287, 1295, 1303, 1311, 1319, 1327

Offset: 1

Reinhard Zumkeller, Jul 21 2012

nonn

A105027(a(n)) = A214417(a(n)) = a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A105271, A214433, A104235.

a214414 n = a214414_list !! (n-1)
a214414_list = [x | x <- [0..], a105027 x == x]

A214417 Inverse permutation to A105027.

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 4, 7, 11, 10, 9, 12, 15, 14, 13, 8, 20, 19, 18, 21, 24, 23, 22, 17, 28, 27, 26, 29, 16, 31, 30, 25, 37, 36, 35, 38, 41, 40, 39, 34, 45, 44, 43, 46, 33, 48, 47, 42, 53, 52, 51, 54, 57, 56, 55, 50, 61, 60, 59, 62, 49, 32, 63, 58, 70, 69, 68

Offset: 0

Reinhard Zumkeller, Jul 21 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A214416, A214414 (fixed points).

-- import Data.List (elemIndex)
-- import Data.Maybe (fromJust)
a214417 = fromJust . (`elemIndex` a105027_list)

A214433 Where A105025 and A105027 agree.

Original entry on oeis.org

0, 1, 2, 3, 9, 10, 13, 14, 34, 37, 42, 45, 50, 53, 58, 61, 517, 522, 533, 538, 549, 554, 565, 570, 581, 586, 597, 602, 613, 618, 629, 634, 645, 650, 661, 666, 677, 682, 693, 698, 709, 714, 725, 730, 741, 746, 757, 762, 773, 778, 789, 794, 805, 810, 821, 826

Offset: 1

Reinhard Zumkeller, Jul 21 2012

A105025(a(n)) = A105027(a(n)).

			A105025(1018) = A105027(1018) = 1011, => 1018 is a term: a(80) = 1018;
A105025(1019) = 994 and A105027(1019) = 1014, => 1019 not a term;
A105025(524302) = A105027(524302) = 524317, => a(81) = 524302;
A105025(524303) = 524312 and A105027(524303) = 524300, => 524302 not a term.
-

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Cf. A214489.

a214433 n = a214433_list !! (n-1)
a214433_list = [x | x <- [0..], a105025 x == a105027 x]

A105154 Consider trajectory of n under repeated application of map k -> A105027(k); a(n) = length of cycle.

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 2, 1, 4, 2, 2, 4, 4, 2, 2, 4, 4, 4, 1, 4, 4, 4, 1, 4, 4, 4, 1, 4, 4, 4, 1, 4, 8, 4, 4, 4, 4, 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 16, 8, 4, 2, 4, 8, 16, 2, 16, 8, 4, 2, 4, 8, 16, 2, 16, 8, 4, 2, 4, 8, 16, 2, 16, 8, 4, 2, 4, 8, 16, 2, 16, 8, 4, 2, 4

Offset: 0

Philippe Deléham, Apr 30 2005

Why is this always a power of 2?

a(n) is always a power of 2: If n is a k-bit number, then so are all numbers in the A105154-orbit of n. For m in the orbit, the i-th bit (i=1,..,k) of A105154(m) is the i-th bit of m-k+i and hence depends only on the lower i bits of m. By induction quickly follows that the lower i bits run through a cycle of length dividing 2^i. This also shows that a(n) <= n for n > 0.

Hagen von Eitzen, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Cf. A102370, A105025, A105027, A105153.

a105154 n = t [n] where
   t xs@(x:_) | y `elem` xs = length xs
              | otherwise   = t (y : xs) where y = a105027 x
-- Reinhard Zumkeller, Jul 21 2012

More terms taken from b-file by Hagen von Eitzen, Jun 24 2009

A105028 Binary equivalents of A105027.

Original entry on oeis.org

0, 1, 11, 10, 110, 101, 100, 111, 1111, 1010, 1001, 1000, 1011, 1110, 1101, 1100, 11100, 10111, 10010, 10001, 10000, 10011, 10110, 10101, 10100, 11111, 11010, 11001, 11000, 11011, 11110, 11101, 111101, 101100, 100111, 100010, 100001

Offset: 0

N. J. A. Sloane, Apr 03 2005

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A007088, A105027.

a(n) = A007088(A105027(n)). - Michel Marcus, May 06 2020

More terms from Philippe Deléham and Erich Friedman, Aug 05 2005

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 3, 6, 5, 4, 15, 10, 9, 8, 11, 14, 13, 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30, 61, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 126, 93, 76, 71, 66, 65, 64, 67, 70, 69

Offset: 0

Philippe Deléham, Feb 13 2005

All terms are distinct, but certain terms (see A102371) are missing. But see A103122.

Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.

			........0
........1
.......10
.......11
......100
......101
......110
......111
.....1000
.........
The upward-sloping diagonals are:
0
11
110
101
100
1111
1010
.......
giving 0, 3, 6, 5, 4, 15, 10, ...
The sequence has a natural decomposition into blocks (see the paper): 0; 3; 6, 5, 4; 15, 10, 9, 8, 11, 14, 13; 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30; 61, ...
Reading the array of binary numbers along diagonals with slope 1 gives this sequence, slope 2 gives A105085, slope 0 gives A001477 and slope -1 gives A105033.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].
Index entries for sequences related to binary expansion of n

Related sequences (1): A103542 (binary version), A102371 (complement), A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, A103582, A103583.
Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 (base 10 version).
Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, A105026, A105027, A105028.
Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, A105035, A105108.
Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, A104490, A104853, A104893, A104894, A105085.

a102370 n = a102370_list !! n
a102370_list = 0 : map (a105027 . toInteger) a062289_list
-- Reinhard Zumkeller, Jul 21 2012

A102370:=proc(n) local t1,l; t1:=n; for l from 1 to n do if n+l mod 2^l = 0 then t1:=t1+2^l; fi; od: t1; end;

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ f[n] + n, {n, 0, 71}] (* Robert G. Wilson v, Mar 21 2005 *)

A102370(n)=n-1+sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k)) \\ Benoit Cloitre, Mar 20 2005

{a(n) = if( n<1, 0, sum( k=0, length( binary( n)), bitand( n + k, 2^k)))} /* Michael Somos, Mar 26 2012 */

def a(n): return 0 if n<1 else sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)]) # Indranil Ghosh, May 03 2017

a(n) = n + Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k. (Cf. A103185.) In particular, a(n) >= n. - N. J. A. Sloane, Mar 18 2005

a(n) = A105027(A062289(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

More terms from Benoit Cloitre, Mar 20 2005

A062289 Numbers n such that n-th row in Pascal triangle contains an even number, i.e., A048967(n) > 0.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001

nonn

Numbers n such that binary representation contains the bit string "10". Union of A043569 and A101082. - Rick L. Shepherd, Nov 29 2004

The asymptotic density of this sequence is 1 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10001
Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016.
Oliver Kullmann and Xishun Zhao, On variables with few occurrences in conjunctive normal forms, in: K. A. Sakallah and L. Simon (eds), International Conference on Theory and Applications of Satisfiability Testing, Springer, Berlin, Heidelberg, 2011, pp. 33-46; arXiv preprint, arXiv:1010.5756 [cs.DM], 2010-2011.
Oliver Kullmann and Xishun Zhao, Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses, arXiv preprint, arXiv:1505.02318 [cs.DM], 2015.

Complement of A000225, so these might be called non-Mersenne numbers.
A132782 is a subsequence.
Cf. A048967, A057716, A043569, A101082.
Cf. A007461, A036987, A102370, A105027.
Cf. A261461, A261922.

a062289 n = a062289_list !! (n-1)
a062289_list = 2 : g 2 where
   g n = nM n : g (n+1)
   nM k = maximum $ map (\i -> i + min i (a062289 $ k-i+1)) [2..k]
   -- Cf. link [Oliver Kullmann, Xishun Zhao], Def. 3.1, page 3.
-- Reinhard Zumkeller, Feb 21 2012, Dec 31 2010

ok[n_] := MatchQ[ IntegerDigits[n, 2], {_, 1, 0, _}]; Select[ Range[100], ok] (* Jean-François Alcover, Dec 12 2011, after Rick L. Shepherd *)

isok(m) = #select(x->((x%2)==0), vector(m+1, k, binomial(m, k-1))); \\ Michel Marcus, Jan 26 2021

def A062289(n): return n+(m:=n.bit_length())-(not n>=(1<Chai Wah Wu, Jun 30 2024

a(n) = A057716(n+1) - 1.
a(n) = 2 if n=1, otherwise max{min{2*i, a(n-i+1) + i}: 1 < i <= n}.
A036987(a(n)) = 0. - Reinhard Zumkeller, Mar 06 2012
A007461(a(n)) mod 2 = 0. - Reinhard Zumkeller, Apr 02 2012
A102370(n) = A105027(a(n)). - Reinhard Zumkeller, Jul 21 2012
A261461(a(n)) = A261922(a(n)). - Reinhard Zumkeller, Sep 17 2015

More terms from Rick L. Shepherd, Nov 29 2004

A102371 Numbers missing from A102370.

Original entry on oeis.org

1, 2, 7, 12, 29, 62, 123, 248, 505, 1018, 2047, 4084, 8181, 16374, 32755, 65520, 131057, 262130, 524279, 1048572, 2097133, 4194286, 8388587, 16777192, 33554409, 67108842, 134217711, 268435428, 536870885

Offset: 1

Philippe Deléham, Feb 13 2005

Indices of negative numbers in A103122.
Write numbers in binary under each other; start at 2^k, read in upward direction with the first bit omitted and convert to decimal:
. . . . . . . . . . 0
. . . . . . . . . . 1
.. . . . . . . . . 10 < -- Starting here, the upward diagonal (first bit omitted) reads 1 -> 1
.. . . . . . . . . 11
. . . . . . . . . 100 < -- Starting here, the upward diagonal (first bit omitted) reads 10 -> 2
. . . . . . . . . 101
. . . . . . . . . 110
. . . . . . . . . 111
.. . . . . . . . 1000 < -- Starting here, the upward diagonal (first bit omitted) reads 111 -> 7
. . . . . . . . .1001
Thus a(n) = A102370(2^n - n) - 2^n.
Do we have a(n) = 2^n-1-A105033(n-1)? - David A. Corneth, May 07 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370, A103530, A103581, A103582, A103583, A105033.

a102371 n = a102371_list !! (n-1)
a102371_list = map (a105027 . toInteger) $ tail a000225_list
-- Reinhard Zumkeller, Jul 21 2012

A102371:= proc (n) local t1, l; t1 := -n; for l to n do if `mod`(n-l,2^l) = 0 then t1 := t1+2^l end if end do; t1 end proc;

a=1
for n in range(2,66):
    print(a, end=",")
    a ^= a+n
# Alex Ratushnyak, Apr 21 2012

a(n) = -n + Sum_{ k >= 1, k == n mod 2^k } 2^k. - N. J. A. Sloane and David Applegate, Mar 22 2005. E.g. a(5) = -5 + 2^1 + 2^5 = 29.
a(2^k + k) -a(k) = 2^(2^k + k) - 2^k, with k>= 1.
a(1)=1, for n>1, a(n) = a(n-1) XOR (a(n-1) + n), where XOR is the bitwise exclusive-or operator. - Alex Ratushnyak, Apr 21 2012
a(n) = A105027(A000225(n)). - Reinhard Zumkeller, Jul 21 2012

More terms from Benoit Cloitre, Mar 20 2005
a(16)-a(22) from Robert G. Wilson v, Mar 21 2005
a(15)-a(29) from David Applegate, Mar 22 2005

A105025 Write numbers in binary under each other; to get the next block of 2^k (k >= 0) terms of the sequence, start at 2^k, read diagonals in downward direction and convert to decimal.

Original entry on oeis.org

0, 1, 3, 2, 4, 7, 6, 5, 11, 10, 9, 12, 15, 14, 13, 8, 18, 17, 20, 23, 22, 21, 16, 27, 26, 25, 28, 31, 30, 29, 24, 19, 33, 36, 39, 38, 37, 32, 43, 42, 41, 44, 47, 46, 45, 40, 35, 50, 49, 52, 55, 54, 53, 48, 59, 58, 57, 60, 63, 62, 61, 56, 51, 34, 68, 71, 70, 69, 64, 75, 74, 73, 76

Offset: 0

N. J. A. Sloane, Apr 03 2005

This is a permutation of the nonnegative integers.

a(A214433(n)) = A105027(A214433(n)); a(A214489(n)) = A105029(A214489(n)). - Reinhard Zumkeller, Jul 21 2012

			........0
........1
.......10
.......11
......100 <- Starting here, the downward diagonals
......101 read 100, 111, 110, 101, giving the block 4, 7, 6, 5.
......110
......111
.....1000
.....1001
.....1010
.....1011
.........

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to binary expansion of n

Cf. A102370, A105026.
Cf. A070939, A000079.
Cf. A105271 (fixed points), A214416 (inverse).

import Data.Bits ((.|.), (.&.))
a105025 n = foldl (.|.) 0 $ zipWith (.&.)
                  a000079_list $ reverse $ enumFromTo n (n - 1 + a070939 n)
-- Reinhard Zumkeller, Jul 21 2012

a:=proc(i,j) if j=1 and i<=16 then 0 else convert(i+15,base,2)[7-j] fi end: seq(a(i,2)*2^4+a(i+1,3)*2^3+a(i+2,4)*2^2+a(i+3,5)*2+a(i+4,6),i=1..16); # this is a Maple program (not necessarily the simplest) only for one block of (2^4) numbers # Emeric Deutsch, Apr 16 2005

numberOfBlocks = 7; bloc[n_] := Join[ Table[ IntegerDigits[k, 2], {k, 2^(n-1), 2^n-1}], Table[ Rest @ IntegerDigits[k, 2], {k, 2^n, 2^n+n}]]; Join[{0, 1}, Flatten[ Table[ Table[ Diagonal[bloc[n], k] // FromDigits[#, 2]&, {k, 0, -2^(n-1)+1, -1}], {n, 2, numberOfBlocks}]]] (* Jean-François Alcover, Nov 03 2016 *)

More terms from Emeric Deutsch, Apr 16 2005

A105029 Write numbers in binary under each other, left justified, read diagonals in downward direction, convert to decimal.

Original entry on oeis.org

0, 2, 6, 5, 4, 14, 13, 8, 11, 10, 9, 12, 30, 29, 24, 19, 18, 17, 20, 23, 22, 21, 16, 27, 26, 25, 28, 62, 61, 56, 51, 34, 33, 36, 39, 38, 37, 32, 43, 42, 41, 44, 47, 46, 45, 40, 35, 50, 49, 52, 55, 54, 53, 48, 59, 58, 57, 60, 126, 125, 120, 115, 98, 65, 68, 71, 70

Offset: 0

Benoit Cloitre, Apr 03 2005

All terms are distinct, but the numbers 2^m - 1 are missing.
a(n) = Sum_{k>=1} B(n+k-1,k)*2^(A103586(n)-k) where B(n,k) n>=1, k>=1 is the infinite array:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
.......
where n-th row consists of binary expansion of n followed by 0's.
a(n) = A105025(n) iff A070939(n) = A103586(n), cf. A214489. - Reinhard Zumkeller, Jul 21 2012

			0
1
1 0
1 1
1 0 0
1 0 1
1 1 0
1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
and reading the diagonals downwards we get 0, 10, 110, 101, 100, 1110, 1101, etc.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences related to binary expansion of n

Cf. A102370, A105030, A105025, A105026, A105027, A105028, A105033.

Cf. A000079, A070939, A103586.

import Data.Bits ((.|.), (.&.))
a105029 n = foldl (.|.) 0 $ zipWith (.&.) a000079_list $
   map (\x -> (len + 1 - a070939 x) * x)
       (reverse $ enumFromTo n (n - 1 + len))  where len = a103586 n
-- Reinhard Zumkeller, Jul 21 2012

cp's OEIS Frontend

A214414 Fixed points of permutations A105027 and A214417.

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A214417 Inverse permutation to A105027.

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A214433 Where A105025 and A105027 agree.

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A105154 Consider trajectory of n under repeated application of map k -> A105027(k); a(n) = length of cycle.

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A105028 Binary equivalents of A105027.

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A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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Maple

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A062289 Numbers n such that n-th row in Pascal triangle contains an even number, i.e., A048967(n) > 0.

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Mathematica

PARI

Python

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A102371 Numbers missing from A102370.

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