A038566 -id:A038566 - cp's OEIS frontend

A339046 Irregular triangle read by rows: row n gives the complete quadrupling system modulo N = 2*n + 1, for n >= 0.

1, 1, 2, 1, 4, 2, 3, 1, 4, 2, 3, 5, 6, 1, 4, 7, 2, 8, 5, 1, 4, 5, 9, 3, 2, 8, 10, 7, 6, 1, 4, 3, 12, 9, 10, 2, 8, 6, 11, 5, 7, 1, 4, 2, 8, 7, 13, 11, 14, 1, 4, 16, 13, 2, 8, 15, 9, 3, 12, 14, 5, 6, 7, 11, 10, 1, 4, 16, 2, 8, 11, 5, 20, 17, 10, 19, 13, 1, 4, 16, 18, 3, 12, 2, 8, 9, 13, 6, 2, 8, 9, 13, 6, 1, 4, 16, 18, 3, 12

Offset: 0

Wolfdieter Lang, Dec 13 2020

The length of row n is given by phi(2*n + 1), with phi = A000010, for n >= 0.
The quadrupling sequence modulo N = 2*n + 1, for n >= 0, has entries QS(N, s(N,i), j) = s(N,i)*4^j (mod N), with j >= 0, and certain positive integer seeds s(N, i), for i = 1, 2, ..., S(N) = A339049((N-1)/2), where gcd(s(N, i), N) = 1 (restricted seeds modulo N). These quadrupling sequences are periodic with period length P(N) = A053447((N-1)/2) (order of 4 modulo N). Only the periods (cycles) QS(N, s(N,i)) = {QS(N, s(N, i), j)}_{j=0..P(N)-1}, for i = 1, 2, ..., S(N), are listed.
N = 1 (n = 0) is special: one takes here the restricted residue system modulo N not as [0] but as [1]. The order of 4 modulo 1 is 1, because 4^1 == 1 (mod 1) (== 0 (mod 1)).
In order to obtain the complete system of quadrupling sequences one starts with seed s(N, 1) = 1, and if all numbers from the smallest positive reduced residue system modulo N (called RRS(N), given in row N of A038566) are obtained, i.e., if P(N) = #RRS(N) = phi(N) = A000010(N), then the system is complete. Otherwise the smallest missing number from RRS(N) is taken as new seed s(N, 2), etc. until the system is complete. This means that the number of seeds needed is S(N) given above.
This entry generalizes A337712, given together with Gary W. Adamson. See also A337936.

			The irregular triangle begins (the vertical bar separates the cycles):
n,  N \ k  1 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 ...
0,  1:     1
1,  3:     1|2
2,  5:     1 4| 2  3
3,  7:     1 4  2| 3  5  6
4,  9:     1 4  7| 2  8  5
5,  11:    1 4  5  9  3| 2  8 10  7  6
6,  13:    1 4  3 12  9 10| 2  8  6 11  5  7
7,  15:    1 4| 2  8| 7 13|11 14
8,  17:    1 4 16 13| 2  8 15  9| 3 12 14  5| 6  7 11 10
9,  19:    1 4 16  7  9 17 11  6  5| 2  8 13 14 18 15  3 12 10
10, 21:    1 4 16| 2  8 11| 5 20 17|10 19 13
11, 23:    1 4 16 18  3 12  2  8  9 13  6| 2  8  9 13  6  1  4 16 18  3 12
12, 25:    1 4 16 14  6 24 21  9 11 19| 2  8  7  3 12 23 17 18 22 13
13, 27:    1 4 16 10 13 25 19 22  7| 2  8  5 20 26 23 11 17 14
...
n = 14, N = 29: 1 4 16 6 24 9 7 28 25 13 23 5 20 22 | 2 8 3 12 19 18 14 27 21 26 17 10 11 15,
n = 15, N = 31: 1 4 16 2 8 | 3 12 17 6 24 | 5 20 18 10 9 | 7 28 19 14 25 | 11 13 21 22 26 | 15 29 23 30 27.
...

Cf. A000010, A053447, A337712 (doubling), A337936 (tripling), A339049.

T(n, k) gives the k-th entry in the complete quadrupling system modulo N = 2*n + 1, for n >= 0, with the S(N) = A339049((N-1)/2) cycles of length A053447((N-1)/2) written in row n. See the comment above for QS(N,s(N,i)), i = 1, 2, ..., S(N).

A381499 a(n) = sum of numbers k < n such that 1 < gcd(k,n) < k and rad(k) does not divide n, where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 0, 10, 0, 28, 28, 42, 0, 39, 0, 65, 65, 80, 0, 102, 45, 126, 96, 159, 0, 111, 0, 210, 148, 210, 138, 253, 0, 280, 221, 338, 0, 342, 0, 411, 366, 444, 0, 547, 140, 563, 403, 601, 0, 700, 344, 708, 512, 750, 0, 751, 0, 868, 703, 930

Offset: 1

Michael De Vlieger, Mar 02 2025

nonn

Analogous to A066760(n), the sum of row n of A133995, and A381497(n), sum of row n of A381094.

			Table of n and a(n) for select n, showing prime power decomposition of the latter and row n of A272619:
 n   a(n)  Factor(a(n))  Row n of A272619
-----------------------------------------------------
 8     6   2 * 3         {6}
 9     6   2 * 3         {6}
10     6   2 * 3         {6}
12    10   2 * 5         {10}
14    28   2^2 * 7       {6,10,12}
15    28   2^2 * 7       {6,10,12}
16    42   2 * 3 * 7     {6,10,12,14}
18    39   3 * 13        {10,14,15}
20    65   5 * 13        {6,12,14,15,18}
21    65   5 * 13        {6,12,14,15,18}
22    80   2^4 * 5       {6,10,12,14,18,20}
24   102   2 * 3 * 17    {10,14,15,20,21,22}
25    45   3^2 * 5       {10,15,20}
26   126   2 * 3^2 * 7   {6,10,12,14,18,20,22,24}
27    96   2^5 * 3       {6,12,15,18,21,24}
28   159   3 * 53        {6,10,12,18,20,21,22,24,26}

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 8..2^14, ignoring a(n) = 0, showing a(n) for prime power n in gold, a(n) for squarefree n in green, otherwise blue.

Cf. A007947, A038566, A066760, A121998, A243823, A272619, A381497.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; If[PrimeQ[n], 0, Total@ Select[Range[n], And[1 < GCD[#, n] < #, ! Divisible[n, rad[#]]] &]], {n, 120}]

a(n) is the sum of row n of A272619.

a(n) = 0 for prime n, n = 4, and n = 6.

A061097 a(n) is the concatenation of the phi(n) numbers between 1 and n that are relatively prime to n.

Original entry on oeis.org

1, 1, 12, 13, 1234, 15, 123456, 1357, 124578, 1379, 12345678910, 15711, 123456789101112, 13591113, 12478111314, 13579111315, 12345678910111213141516, 157111317, 123456789101112131415161718

Offset: 1

Amarnath Murthy, Apr 19 2001

			a(6) = 15, 1 and 5 are the two coprime numbers less than 6.
a(7) = 123456. 7 is a prime. phi(7) = 6 hence all the numbers less than 7 are concatenated.

Robert Israel, Table of n, a(n) for n = 1..372

Cf. A000010, A038566.

f:= proc(n) local i,t;
  t:= 1;
  for i from 2 to n-1 do if igcd(i,n)=1 then t:= t*10^(1+ilog10(i))+i fi od;
  t
end proc:
map(f, [$1..30]); # Robert Israel, Jun 13 2018

a(n)=j=0;for(k=1,n,if(gcd(k,n)==1,j=j*10^#digits(k)+k));j \\ Eric Chen, Jun 13 2018

a(n) = eval(concat(apply(x->Str(x), select(x->(gcd(n, x) == 1), [1..n])))); \\ Michel Marcus, Jun 14 2018

a(11)-a(19) from Carl Najafi, Apr 30 2011

A092249 Positions of the integers in the standard diagonal enumeration of the rationals (with the integers in the first column and diagonals moving up to the right).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964

Offset: 1

Andrew Niedermaier, Feb 20 2004

nonn

A002088 without the leading zero. [R. J. Mathar, Jul 20 2009]

			The first few terms of the full enumeration are 1, 2, 1/2, 3, 1/3, 4, 3/2, 2/3, 1/4, 5, giving a(n) = 1, 2, 4, 6, 10,...
Contribution from _R. J. Mathar_, Jul 20 2009: (Start)
The positions in the first column of the table
....1..1/2..1/3..1/4..1/5..1/6..1/7..1/8..1/9.1/10.1/11.1/12
....2.......2/3.......2/5.......2/7.......2/9......2/11.....
....3..3/2.......3/4..3/5.......3/7..3/8......3/10.3/11.....
....4.......4/3.......4/5.......4/7.......4/9......4/11.....
....5..5/2..5/3..5/4.......5/6..5/7..5/8..5/9......5/11.5/12
....6.................6/5.......6/7................6/11.....
....7..7/2..7/3..7/4..7/5..7/6.......7/8..7/9.7/10.7/11.7/12
....8.......8/3.......8/5.......8/7.......8/9......8/11.....
....9..9/2.......9/4..9/5.......9/7..9/8......9/10.9/11.....
...10......10/3................10/7......10/9.....10/11.....
...11.11/2.11/3.11/4.11/5.11/6.11/7.11/8.11/911/10.....11/12
...12................12/5......12/7...............12/11.....
if scanned along rising antidiagonals, as defined by the ratios A038566(i)/A020653(i). (End)

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A000010, A002088, A020653, A038566.

Accumulate[EulerPhi[Range[100]]] (* Paolo Xausa, Oct 19 2023 *)

a(11) and a(12) corrected by R. J. Mathar, Jul 20 2009
Incorrect recurrence formula removed by R. J. Mathar, Jul 29 2009
More terms (using A002088) from Michel Marcus, Sep 10 2018

A128248 a(n) = Sum_{k=1..phi(n)} t(k,n)*(-1)^k, where t(k,n) is the k-th positive integer that is coprime to n and phi(n) = A000010(n).

Original entry on oeis.org

-1, -1, 1, 2, 2, 4, 3, 4, 3, 4, 5, 8, 6, 8, 8, 8, 8, 12, 9, 8, 8, 12, 11, 16, 10, 12, 9, 16, 14, 16, 15, 16, 16, 16, 16, 24, 18, 20, 16, 16, 20, 16, 21, 24, 24, 24, 23, 32, 21, 20, 24, 24, 26, 36, 24, 32, 24, 28, 29, 32, 30, 32, 24, 32, 32, 32, 33, 32, 32, 32, 35, 48, 36, 36, 40, 40, 40, 32, 39, 32, 27, 40, 41, 32, 40, 44, 40, 48, 44, 48, 48

Offset: 1

Leroy Quet, May 03 2007

sign

a(1) and a(2) are the only negative terms of the sequence.

			The positive integers which are <= 10 and are coprime to 10 are 1,3,7,9. So a(10) = -1 + 3 - 7 + 9 = 4.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000010, A038566.

with(numtheory): t:=proc(k,n) local A,i: A:={}: for i from 1 while nops(A)<=k do if igcd(i,n)=1 then A:=A union {i} else A:=A: fi od: A[k] end: a:=n->add((-1)^k*t(k,n),k=1..phi(n)): seq(a(n),n=1..100); # Emeric Deutsch, May 06 2007

Table[Total[Times@@@Partition[Riffle[Select[Range[n],CoprimeQ[#,n]&],{-1,1},{2,-1,2}],2]],{n,100}] (* Harvey P. Dale, May 05 2013 *)

More terms from Emeric Deutsch, May 06 2007

A141455 Irregular triangle showing the set of all possible values of primes modulo n in row n.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 5, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 5, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 5, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 5, 7, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 5, 7, 9, 11, 13, 1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 1, 2, 3, 5, 7, 9, 11, 13, 15

Offset: 2

Roger L. Bagula and Gary W. Adamson, Aug 07 2008

			Table begins:0, 1;
0, 1, 2;
1, 2, 3;
0, 1, 2, 3, 4;
1, 2, 3, 5;
0, 1, 2, 3, 4, 5, 6;
1, 2, 3, 5, 7;
1, 2, 3, 4, 5, 7, 8;
1, 2, 3, 5, 7, 9;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
1, 2, 3, 5, 7, 11;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
1, 2, 3, 5, 7, 9, 11, 13;
1, 2, 3, 4, 5, 7, 8, 11, 13, 14;
1, 2, 3, 5, 7, 9, 11, 13, 15;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
1, 2, 3, 5, 7, 11, 13, 17;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18;
1, 2, 3, 5, 7, 9, 11, 13, 17, 19;

Michael De Vlieger, Table of n, a(n) for n = 2..10209 (rows n = 2..180, flattened)

Cf. A057859 (row lengths), A039701 (row n=3), A039704 (row n=6), A027748, A038566.

Table[Union[FactorInteger[n][[All, 1]] /. n -> 0, Select[Range[n - 1], CoprimeQ[n, #] &]], {n, 2, 15}] (* Michael De Vlieger, Apr 18 2022 *)

Row n = A027748(n) U A038566(n), writing n as 0 iff n is prime. - Michael De Vlieger, Apr 18 2022

A216251 a(n) = n-th decimal digit of the decimal expansion of the n-th Farey fraction ordered by rank.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Mar 14 2013

Used as an example in the Schaffter link to support Cantor's diagonal argument. Most probably irrational and possible normal.

			The first decimal digit of 0 is 0, the second decimal digit of 1 (=0.99999...) is 9, the third decimal digit of 1/2 is 0, the fourth decimal digit of 1/3 is 3, the fifth decimal digit of 2/3 is 6, ..., the twelfth decimal digit of 1/6 is 6, the thirteenth decimal digit of 5/6 is 3, the fourteenth decimaldigit of 1/7 is 4, ..., .

Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin Heidelberg NY, Section of Analysis, Chptr 15, "Sets, function, and the continuum hypothesis", 2000, pp. 87 - 98.
Georg Cantor, Über eine Eigenschaft des Inbegriffes aller reellen Zahlen ("On the Characteristic Property of All Real Numbers")
Timothy Gowers, Editor, with June Barrow-Green & Imre Leader, Assc. Editors, The Princeton Companion to Mathematics, Princeton Un. Press, Princeton & Oxford, 2008, pp. 171 & 779
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §7.5 Transfinite Numbers, pp. 257-262.

Richard Lipton, Gödel's Lost Letter and P=NP
Luke Mastin, 19th Century Mathematics - Cantor
Tom Schaffter, Cantor's Diagonal Argument: Proof and Paradox
Eric Weisstein's World of Mathematics, Cantor Diagonal Method
Wikipedia, Cantor's diagonal argument

Cf. A071989, A038566, A038567.

FareyOrder[n_] := Select[ Table[a/n, {a, n}], Denominator[#] == n &]; lst = Join[{0, .999999}, Flatten[ Table[ FareyOrder[n], {n, 2, 19}]]]; f[n_] := RealDigits[ lst[[n]], 10, 2 n][[1, n]]; Array[f, 105]

The n-th decimal digit of the n-th Farey fraction in order, i.e., 0, 1 (=0.99999...), 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, ..., . (this is A038566/A038567)

A224537 Least odd number d such that the Collatz iteration of d has the following property: if the length of the iteration is b and the maximum value occurs at c, the ratio c/b is the n-th Farey-like fraction.

Original entry on oeis.org

1, 3, 5, 215, 21, 1625, 13, 901, 467, 771667051

Offset: 1

T. D. Noe, Apr 23 2013

nonn

The Farey-like fractions are 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6,.... The numerators/denominators are in A038566/A038567. We conclude (unsurprisingly) that the maximum value of a Collatz iteration usually appears toward the beginning of the iteration.

			Viewed as an irregular triangle, with 0's for unknown terms:
1
3,
5, 215,
21, 1625,
13, 901, 467, 771667051,
53, 0,
67, 35, 1815, 43641, 97, 0,
141, 23, 90435, 0,
93, 79, 15, 895, 763, 0,
61, 603, 1224211, 0,
37, 267, 59, 8857, 5793, 335737, 60221, 5434799, 0, 0

Cf. A224533.

A278338 Irregular triangle read by rows in which row n contains the first Carmichael number equal to m mod n where m is coprime to n, 0 <= m < n, ordered by m.

Original entry on oeis.org

561, 561, 1105, 2465, 561, 8911, 561, 46657, 52633, 1729, 1105, 2465, 561, 46657, 294409, 29341, 512461, 1105, 561, 1024651, 2821, 8911, 1729, 1909001, 2821, 162401, 1105, 2465, 561, 52633, 46657, 1729, 2465, 1729, 10585, 29341, 1105, 46657, 1193221

Offset: 1

Charles R Greathouse IV, Nov 18 2016

The n-th row contains phi(n) terms. Wright proves that this sequence exists for each coprime m and n.

			561 = 0 mod 1;
561 = 1 mod 2;
1105 = 1 mod 3, 2465 = 2 mod 3;
561 = 1 mod 4, 8911 = 3 mod 4;
561 = 1 mod 5, 46657 = 2 mod 5, 52633 = 3 mod 5, 1729 = 4 mod 5;
1105 = 1 mod 6, 2465 = 5 mod 6;

Charles R Greathouse IV, Rows n = 1..179 of triangle, flattened
Thomas Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bulletin of the London Mathematical Society 45:5 (2013), pp. 943-952.

Cf. A002997, A038566, A038567.

a(n) is the least Carmichael number equal to A038566(n) mod A038567(n).

A285788 Irregular triangle T(n,m): nonprime 1 <= k <= n such that n and k are coprime.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 1, 4, 8, 1, 9, 1, 4, 6, 8, 9, 10, 1, 1, 4, 6, 8, 9, 10, 12, 1, 9, 1, 4, 8, 14, 1, 9, 15, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 1, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 1, 9, 1, 4, 8, 10, 16, 20, 1, 9, 15, 21, 1, 4, 6, 8, 9, 10

Offset: 1

Michael De Vlieger, Apr 26 2017

Row n is a subset of A038566(n) such that the union of a(n) and A112484(n) = A038566(n).
Row lengths are A048864(n) = A000010(n)-(A000720(n)-A001221(n)), i.e., phi(n)-(pi(n)-omega(n)).
1 appears in every row since 1 is not prime and coprime to all n.
4 is the smallest composite and appears first in row 5 since 4 divides 4.
Rows that contain the single term 1 are in A048597; the largest n = 30 such that the only term is 1.
For prime p, row p contains 1 and all composites k < p, since 1 < m < p are coprime to p.

			Triangle begins:
  n\m  1  2   3   4  5   6   7
   1:  1
   2:  1
   3:  1
   4:  1
   5:  1  4
   6:  1
   7:  1  4   6
   8:  1
   9:  1  4   8
  10:  1  9
  11:  1  4   6   8  9  10
  12:  1
  13:  1  4   6   8  9  10  12
  14:  1  9
  15:  1  4   8  14
  16:  1  9  15
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11055 (rows 1 <= n <= 240)

Cf. A038566, A048597, A048864, A112484.

Table[Select[Range@ n, And[! PrimeQ@ #, CoprimeQ[#, n]] &], {n, 23}] // Flatten

from sympy import gcd, isprime
def a(n): return list(filter(lambda k: isprime(k)==0 and gcd(k, n)==1, range(1, n + 1)))
for n in range(1, 21): print(a(n)) # Indranil Ghosh, Apr 26 2017

cp's OEIS Frontend

A339046 Irregular triangle read by rows: row n gives the complete quadrupling system modulo N = 2*n + 1, for n >= 0.

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A381499 a(n) = sum of numbers k < n such that 1 < gcd(k,n) < k and rad(k) does not divide n, where rad = A007947.

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A061097 a(n) is the concatenation of the phi(n) numbers between 1 and n that are relatively prime to n.

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A092249 Positions of the integers in the standard diagonal enumeration of the rationals (with the integers in the first column and diagonals moving up to the right).

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A128248 a(n) = Sum_{k=1..phi(n)} t(k,n)*(-1)^k, where t(k,n) is the k-th positive integer that is coprime to n and phi(n) = A000010(n).

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A141455 Irregular triangle showing the set of all possible values of primes modulo n in row n.

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A216251 a(n) = n-th decimal digit of the decimal expansion of the n-th Farey fraction ordered by rank.

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A224537 Least odd number d such that the Collatz iteration of d has the following property: if the length of the iteration is b and the maximum value occurs at c, the ratio c/b is the n-th Farey-like fraction.

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