Frank M Jackson - cp's OEIS frontend

A386980 Number of acute Heronian triangles with integer inradius n.

0, 0, 1, 1, 0, 4, 0, 2, 2, 2, 0, 6, 0, 1, 4, 3, 0, 8, 0, 6, 7, 2, 0, 17, 1, 0, 2, 8, 0, 14, 0, 3, 6, 1, 4, 17, 0, 0, 4, 12, 0, 27, 0, 4, 13, 1, 0, 27, 1, 4, 2, 4, 0, 13, 5, 14, 2, 0, 0, 32, 0, 0, 14, 4, 3, 18, 0, 5, 3, 15, 0, 41, 0, 0, 10, 4, 7, 16, 0, 18, 3, 0, 0, 60, 2, 0, 2, 18, 0, 39, 9

Offset: 1

Frank M Jackson, Aug 11 2025

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is acute iff n < (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5). Also, it has been proved that other than n = 3, all acute Heronian triangles have no prime inradii. For n = 3, the Heronian triangle has sides (10, 10, 12).
Empirically, it appears that the remaining occurrences of zero counts (other than 1 and the primes excluding 3) are inradii of the form 2p where p is in the set 13, 19, 29 and all other primes > 29.
The number of right integer triangles with inradius n is given by A078644, the number of obtuse Heronian triangles with inradius n is given by A386981 and the total number of Heronian triangles with inradius n is given by A120062.

			a(6) = 4, and the 4 acute Heronian triangles with inradius 6 have sides (15, 34, 35), (17, 25, 28), (17, 25, 26), (20, 20, 24).

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
Frank M Jackson, Mathematica program

Cf. A078644, A120062, A386981.

Mathematica
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(* See link above. *)
```

A386981 Number of obtuse Heronian triangles with integer inradius n.

Original entry on oeis.org

0, 3, 9, 14, 12, 35, 21, 39, 44, 44, 23, 124, 28, 73, 97, 81, 30, 166, 31, 130, 169, 95, 39, 283, 59, 90, 131, 208, 33, 347, 43, 160, 196, 109, 160, 466, 35, 117, 197, 304, 41, 515, 57, 267, 354, 127, 61, 550, 110, 214, 219, 258, 44, 425, 215, 484, 265, 128, 51, 977, 41, 138, 582, 269, 169, 603, 48, 325, 252, 564, 47, 1058, 65, 133, 445, 341

Offset: 1

Frank M Jackson, Aug 11 2025

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is obtuse iff n > (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5).
The number of right integer triangles with inradius n is given by A078644, the number of acute Heronian triangles with inradius n is given by A386980 and the total number of Heronian triangles with inradius n is given by A120062.

			a(2) = 3, and the 3 obtuse Heronian triangles with inradius 2 have sides (6, 25, 29), (7, 15, 20), (9, 10, 17).

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
Frank M Jackson, Mathematica program

Cf. A078644, A120062, A386980

Mathematica
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(* See link above. *)
```

A385916 Positive integers m that form Gaussian integers m + i such that every Gaussian integer g with |g| <= |m + i| is a linear combination of the distinct Gaussian divisors of m + i (where i is the imaginary unit).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 12, 13, 17, 18, 21, 23, 27, 31, 32, 33, 37, 38, 41, 43, 47, 55, 57, 68, 72, 73, 75, 81, 82, 83, 89, 91, 93, 98, 99, 105

Offset: 1

Frank M Jackson, Jul 12 2025

This sequence is an example of extending the concept of a practical number to the domain of Gaussian integers. To determine if a Gaussian integer p is practical over the Gaussian integer domain it is necessary to show that the Gaussian divisors (including all their associates) of the Gaussian integer p when combined linearly and distinctly generate all Gaussian integers g where |g| <= |p|.
The Mathematica program in the link below gives a complex plot of the linear combinations of the distinct divisors of a Gaussian integer m + i to see if it is a member of this sequence.
An analogous sequence such that positive integers m that form the Gaussian integers m + i are prime is given by A005574.

			a(3) is in the sequence because the Gaussian divisors of 3 + i are 1, 1 + i, 1 + 2i, 3 + i. Each divisor has 3 other associates. In total these 16 divisors will give the complex plot below when they are combined linearly and distinctly. Note that the patten in any quadrant is a rotation by a right angle of its adjacent quadrant.
|= = = = = = = = = = + = = = = = = = = = =|
|                    * * *                |
|              * * * * * * * *            |
|        * * * * * * * * * * * * *        |
|      * * * * * * * * * * * * * * *      |
|    * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * @ @ @ * * * * * * * *  |
|* * * * * * * * @ @ @ @ @ * * * * * * *  |
|* * * * * * * @ @ @ @ @ @ @ * * * * * *  |
+*-*-*-*-*-*-*-@-@-@-@-@-@-@-*-*-*-*-*-*-*+
|  * * * * * * @ @ @ @ @ @ @ * * * * * * *|
|  * * * * * * * @ @ @ @ @ * * * * * * * *|
|    * * * * * * * @ @ @ * * * * * * * *  |
|    * * * * * * * * * * * * * * * * * *  |
|    * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * *    |
|      * * * * * * * * * * * * * * *      |
|        * * * * * * * * * * * * *        |
|            * * * * * * * *              |
|                * * *                    |
|= = = = = = = = = = + = = = = = = = = = =|

Frank M Jackson, Mathematica program that gives a complex plot

Cf. A005574, A005153, A363227, A385489.

A385489 Positive integers m such that every Gaussian integer g with |g| <= m is a linear combination of the distinct Gaussian divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 55, 56, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 95, 96, 98, 99, 100

Offset: 1

Frank M Jackson, Jun 30 2025

nonn

Practical numbers (A005153) are defined over Z+. A generalization of practical numbers over Z are known as "semi-practical" numbers (A363227). This sequence is a further generalization over the Gaussian integers.
It is assumed that all "semi-practical" numbers are members of this sequence.
The Mathematica program in the link below gives a complex plot of the linear combinations of the distinct divisors of a positive integer to see if it is a member of this sequence.

			a(5) is in the sequence because the Gaussian divisors of 5 are 1, 1+2i, 2+i, 5. Each divisor has 3 other associates. In total these 16 divisors will give the complex plot below when they are combined linearly and distinctly. 5 is not a "semi-practical" number. Note also that every similar complex plot will give a pattern with the same number of axes of symmetry as that of a square.
|= = = = = = = = = = = = + = = = = = = = = = = = =|
|            * * *     * * *     * * *            |
|        * * * * * * * * * * * * * * * * *        |
|    * * * * * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|* * * * * * * * * * * * * * * * * * * * * * * * *|
|* * * * * * * * * * * * @ * * * * * * * * * * * *|
|* * * * * * * * * @ @ @ @ @ @ @ * * * * * * * * *|
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|* * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * * *|
+*-*-*-*-*-*-*-@-@-@-@-@-@-@-@-@-@-@-*-*-*-*-*-*-*+
|* * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * * *|
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|* * * * * * * * * @ @ @ @ @ @ @ * * * * * * * * *|
|* * * * * * * * * * * * @ * * * * * * * * * * * *|
|* * * * * * * * * * * * * * * * * * * * * * * * *|
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|    * * * * * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * * * * * *    |
|        * * * * * * * * * * * * * * * * *        |
|            * * *     * * *     * * *            |
|= = = = = = = = = = = = + = = = = = = = = = = = =|

Frank M Jackson, Mathematica program that gives a complex plot

Cf. A005153, A363227.

A380328 2-dense squarefree numbers: Squarefree numbers whose divisors increase by factors of at most 2.

Original entry on oeis.org

1, 2, 6, 30, 42, 66, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1110, 1122, 1218, 1230, 1254, 1290, 1302, 1410, 1518, 1554, 1590, 1722, 1770, 1806, 1914, 1974, 2046, 2226, 2310, 2442, 2478, 2562, 2706, 2730, 2814, 2838, 2982, 3066, 3102, 3318, 3486, 3498

Offset: 1

Frank M Jackson, Jan 21 2025

nonn

This sequence is a subsequence of primitive practical numbers (A267124) because the sequence of 2-dense numbers (A174973) is a subsequence of practical numbers (A005153) and all squarefree practical numbers (A265501) are by definition primitive practical numbers.
Similar to and a subsequence of A265501.
Let N(x) be the number of terms less than x. Saias (1997) showed that N(x) has order of magnitude x/log(x). We have N(x) = c*x/log(x) + O(x/(log(x))^2), where c=0.06864... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 14.56... (see Weingartner (2019)). - Andreas Weingartner, Jan 23 2025

			a(5) = 42 and its prime factorization is 2*3*7 and squarefree. Also the proper divisors are 1, 2, 3, 6, 7, 21, 42 they are 2-dense and therefore 42 is practical as well as being primitive practical.

Frank M Jackson, Table of n, a(n) for n = 1..10000
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.

Intersection of A005117 and A174973.
Subsequence of A267124 and of A265501.
Cf. A005153.

filter:= proc(n) local D,i;
  if not numtheory:-issqrfree(n) then return false fi;
  D:= sort(convert(numtheory:-divisors(n),list));
  andmap(i -> D[i+1]<=2*D[i],[$1..nops(D)-1])
end proc:
select(filter, [1,seq(i,i=2..5000,4)]); # Robert Israel, Jan 23 2025

Dens2DivQ[n_] := Module[{lst=Divisors[n], m, ok}, If[n==1, Return[True]]; Do[ok=False; If[lst[[m+1]]/lst[[m]]>2, Break[]]; ok=True, {m, 1, Length[lst]-1}]; ok]; Select[Range[10000], SquareFreeQ[#]&&Dens2DivQ[#]&]

a(n) = C*n*log(n*log(n)) + O(n), where C = 14.56... (see comments). - Andreas Weingartner, Jan 23 2025

A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 20, 16, 18, 40, 28, 32, 24, 80, 56, 30, 64, 36, 100, 112, 60, 42, 128, 48, 160, 196, 90, 84, 66, 256, 54, 200, 224, 120, 126, 132, 78, 512, 72, 320, 392, 150, 168, 198, 156, 88, 1024, 96, 400, 448, 180, 252, 264, 234, 176, 104, 2048, 108, 500, 784, 240, 294, 396, 312, 352, 208, 140, 4096, 144, 640, 896, 270, 336, 528, 468, 704, 416, 280, 204, 8192

Offset: 1

Frank M Jackson, Dec 30 2024

A permutation of the practical numbers.
This sequence is presented as an array of rows. The first row contains a single term of value 1. Subsequent rows are infinite sequences and are presented as a square array by listing the antidiagonals downwards that is 1: 2; 4,6; 8,12,20; etc.
The first column contains the primitive practical numbers A267124; each row lists all practical numbers (A005153) having the same primitive practicle progenitor and which is the first term in each row. See A379325 comments for further details. If T[1,m] is squarefree then the row is identical to the same squarefree row in A284457.
Every primitive practical number A267124(n) is the progenitor of a disjoint subsequence of the practical numbers. If the PP column represents the sequence of primitive practical numbers A267124, the table below give the 7 initial terms of the disjoint sequences of practical numbers A005153 generated by the initial 7 terms of the sequence of primitive practical numbers.
PP: Disjoint subsequence of A005153
--  -------------------------------
1:   1
2:   2,  4,  8, 16, 32, 64,128, . . .- A000079 with offset 1,1
6:   6, 12, 18, 24, 36, 48, 54, . . .- A033845
20: 20, 40, 80,100,160,200,320, . . .
28: 28, 56,112,196,224,392,448, . . .
30: 30, 60, 90,120,150,180,240, . . .- A143207
42: 42, 84,126,168,252,294,336
...
Row 1 is T[1,1] = 1 and only has one term in the subsequence.
Row 7 is T[1,7] = 2*3*7; T[2,7] = 2^2*3*7; T[3,7] = 2*3^2*7; T[4,7] = 2^3*3*7; T[5,7] = 2^2*3^2*7, etc.

			a(14) = 80 and it is T[3,4] = 2^4*5. Its primitive progenitor is 20 = 2^2*5 and its equivalence class are the terms of row 4.

Frank M Jackson, Mathematica program

Cf. A000079, A005117, A005153, A007947, A033845, A143207, A267124, A284457, A379325.

Mathematica
```
(* See link above *)
```

A379698 Exponent of highest power of 2 that divides the n-th practical number A005153(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 2, 1, 5, 2, 3, 1, 4, 1, 3, 2, 6, 1, 3, 1, 4, 2, 3, 1, 5, 2, 3, 2, 4, 3, 1, 7, 2, 2, 4, 1, 2, 5, 1, 3, 4, 2, 6, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 1, 4, 2, 8, 2, 3, 1, 4, 2, 3, 5, 1, 2, 4, 1, 2, 3, 6, 2, 1, 4, 2, 1, 2, 5, 3, 2, 4, 1, 2, 7, 1, 3, 2, 4, 3, 1, 5, 2, 4, 3, 6, 1, 3, 2, 1, 4, 2, 2, 5, 1, 4, 2, 3, 1, 9, 3, 1, 4, 2, 2, 5, 1

Offset: 1

Frank M Jackson, Dec 30 2024

nonn

The relationship between a practical number and highest power of 2 that divides it provides an equivalence relation. Practical numbers whose prime factorization have the same exponent of 2.

			a(9) = 2. A005153(9) = 20 and the greatest power of 2 that divides 20 is 2.

Frank M Jackson, Table of n, a(n) for n = 1..10000

Cf. A005153, A007814, A379325.

plst=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; lst={}; Do[If[OddQ[plst[[n]]], AppendTo[lst, 0], AppendTo[lst, Last@First@FactorInteger[plst[[n]]]]], {n, 1, 100}]; lst

a(n) = A007814(A005153(n)).

A379325 Largest primitive practical number p that divides the n-th practical number - A005153(n) such that the radical of the quotient A005153(n)/p is a divisor of p.

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 6, 20, 6, 28, 30, 2, 6, 20, 42, 6, 6, 28, 30, 2, 66, 6, 78, 20, 42, 88, 30, 6, 20, 104, 6, 28, 30, 42, 2, 66, 140, 6, 30, 78, 20, 6, 42, 88, 30, 6, 28, 66, 20, 204, 104, 210, 6, 220, 28, 228, 78, 30, 42, 2, 260, 66, 30, 272, 276, 140, 6, 42, 30, 304, 306, 308, 78, 20, 6, 330, 42, 340, 342, 348, 88, 30, 364, 368, 42, 380, 6, 390, 28, 66

Offset: 1

Frank M Jackson, Dec 20 2024

nonn

The relationship between a practical number and the largest primitive practical number that divides it such that the radical of the quotient is a divisor of same primitive provides an equivalence relation. Practical numbers that have the same progenitive primitive. This property arises from the characteristic of practical numbers and, in particular, primitives that says a practical number multiplied by power combinations of any of its divisors is also practical. This sequence identifies the primitive progenitor of each practical number A005153(n).

Note that this sequence and A378202 are similar but the first difference is at a(63) as explained in the example.

			a(63) = 66. A005153(63) = 264 and the largest primitive practical number that divides the practical number 264 is 88. However the radical of the quotient 264/88 is 3 and 3 is not a divisor of 88. The next greatest primitive divisor of 264 is 66 and the radical of the quotient 264/66 is 2 and 2 is a divisor of 66.
a(131) = 306. A005153(131) = 612 and it is divisible by two primitive practical numbers 204 and 306 with their quotient a divisor of their primitive in both cases but 306 is chosen as the larger primitive.

Frank M Jackson, Table of n, a(n) for n = 1..10000

Cf. A005153, A007947, A267124, A377377.

plst=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; pplst=Last/@ReadList["https://oeis.org/A267124/b267124.txt", {Number, Number}]; Rad[n_] := Times @@ First /@ FactorInteger[n]; getpplst[n_] := Module[{}, Select[pplst, #<=n &]]; lst1={}; Do[lst=getpplst[plst[[n]]]; lnh=Length@lst; m=0; While[Mod[j=plst[[n]], k=lst[[lnh-m]]]!=0||Mod[k, Rad[j/k]]!=0, m++]; AppendTo[lst1, {j, k}], {n, 1, 100}]; Last/@lst1

A378729 Characteristic function of the primitive practical numbers as a subsequence of the practical numbers.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1

Offset: 1

Frank M Jackson, Dec 06 2024

nonn

a(n) = 1 indicates that the practical number A005153(n) is also primitive.

			a(9) = 1 because A005153(9) = 20 and 20 is primitive practical.

Cf. A005153, A267124.

plst=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; pplst=Last/@ReadList["https://oeis.org/A267124/b267124.txt", {Number, Number}]; getpplst[n_] := Module[{}, Select[pplst, # <= n &]]; Table[lst=getpplst[plst[[n]]]; If[MemberQ[lst, plst[[n]]], 1, 0], {n, 1, 200}]

A378202 Largest primitive practical number that divides the n-th practical number - A005153(n).

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 6, 20, 6, 28, 30, 2, 6, 20, 42, 6, 6, 28, 30, 2, 66, 6, 78, 20, 42, 88, 30, 6, 20, 104, 6, 28, 30, 42, 2, 66, 140, 6, 30, 78, 20, 6, 42, 88, 30, 6, 28, 66, 20, 204, 104, 210, 6, 220, 28, 228, 78, 30, 42, 2, 260, 88, 30, 272, 276, 140, 6, 42, 30, 304, 306, 308, 104, 20, 6, 330, 42, 340, 342, 348, 88, 30, 364, 368, 42, 380, 6, 390, 28, 66

Offset: 1

Frank M Jackson, Nov 19 2024

nonn

Every practical number has a primitive practical divisor. a(n) is the largest primitive practical number that divides A005153(n). The quotient of A005153(n)/a(n) is A377377(n).

			a(8) = 6 because A005153(8) = 18 and the largest primitive practical that divides 18 is 6.

Cf. A005153, A267124, A377377.

plst=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; pplst=Last/@ReadList["https://oeis.org/A267124/b267124.txt", {Number, Number}]; getpplst[n_] := Module[{}, Select[pplst, #<=n &]]; lst1={}; Do[lst=getpplst[plst[[n]]]; lnh=Length@lst; m = 0; While[!IntegerQ[j=plst[[n]]/(k=lst[[lnh-m]])], m++]; AppendTo[lst1, {k, j}], {n, 1, 100}]; First/@lst1

cp's OEIS Frontend

User: Frank M Jackson

A386980 Number of acute Heronian triangles with integer inradius n.

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Mathematica

A386981 Number of obtuse Heronian triangles with integer inradius n.

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Mathematica

A385916 Positive integers m that form Gaussian integers m + i such that every Gaussian integer g with |g| <= |m + i| is a linear combination of the distinct Gaussian divisors of m + i (where i is the imaginary unit).

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A385489 Positive integers m such that every Gaussian integer g with |g| <= m is a linear combination of the distinct Gaussian divisors of m.

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A380328 2-dense squarefree numbers: Squarefree numbers whose divisors increase by factors of at most 2.

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Maple

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A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).

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Mathematica

A379698 Exponent of highest power of 2 that divides the n-th practical number A005153(n).

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Mathematica

Formula

A379325 Largest primitive practical number p that divides the n-th practical number - A005153(n) such that the radical of the quotient A005153(n)/p is a divisor of p.

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