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A006842 Triangle read by rows: row n gives numerators of Farey series of order n.
(Formerly M0041)
51
0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 5, 3, 4, 5, 6, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 5, 6, 7, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 4, 1, 5, 4, 3, 5, 2, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923. See Vol. 1.
Guthery, Scott B. A motif of mathematics. Docent Press, 2011.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
A. O. Matveev, Farey Sequences, De Gruyter, 2017.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Maxim Bruckheimer and Abraham Arcavi, Farey series and Pick’s area theorem, The Mathematical Intelligencer, 17.4 (1995): 64-67.
Cristian Cobeli and Alexandru Zaharescu, The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform 5 (2003): 1-38.
Andrey O. Matveev, Neighboring Fractions in Farey Subsequences, arXiv:0801.1981 [math.NT], 2008-2010.
Eric Weisstein's World of Mathematics, Farey Sequence.
EXAMPLE
0/1, 1/1;
0/1, 1/2, 1/1;
0/1, 1/3, 1/2, 2/3, 1/1;
0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1;
0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1;
MAPLE
Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end: seq(numer(Farey(i)), i=1..5); # Peter Luschny, Apr 28 2009
MATHEMATICA
Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Flatten[ Table[ Numerator[ Farey[n]], {n, 0, 9}]] (* Robert G. Wilson v, Apr 08 2004 *)
Table[FareySequence[n] // Numerator, {n, 1, 9}] // Flatten (* Jean-François Alcover, Sep 25 2018 *)
PROG
(PARI) row(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k); ); ); vf = vecsort(Set(vf)); for (i=1, #vf, print1(numerator(vf[i]), ", ")); } \\ Michel Marcus, Jun 27 2014
CROSSREFS
Row n has A005728(n) terms. - Michel Marcus, Jun 27 2014
Cf. A006843 (denominators), A049455, A049456, A007305, A007306. Also A177405/A177407.
Sequence in context: A054635 A003137 A353048 * A299038 A273693 A219967
KEYWORD
nonn,nice,frac,tabf
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Apr 08 2004
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)