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A006843
Triangle read by rows: row n gives denominators of Farey series of order n.
(Formerly M0081)
62
1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 3, 2, 3, 4, 1, 1, 5, 4, 3, 5, 2, 5, 3, 4, 5, 1, 1, 6, 5, 4, 3, 5, 2, 5, 3, 4, 5, 6, 1, 1, 7, 6, 5, 4, 7, 3, 5, 7, 2, 7, 5, 3, 7, 4, 5, 6, 7, 1, 1, 8, 7, 6, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 6, 7, 8, 1, 1, 9, 8, 7, 6, 5, 9, 4, 7, 3, 8, 5, 7, 9, 2, 9, 7, 5, 8, 3, 7
OFFSET
1,4
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152.
Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 199.
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
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
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(denom(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[ Denominator[ Farey[n]], {n, 9}]] (* Robert G. Wilson v, Apr 08 2004 *)
Table[Denominator[FareySequence[n]], {n, 10}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 04 2016 *)
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(denominator(vf[i]), ", ")); } \\ Michel Marcus, Jun 27 2014
CROSSREFS
Row n has A005728(n) terms. - Michel Marcus, Jun 27 2014
Row sums give A240877.
Cf. A006842 (numerators), A049455, A049456, A007305, A007306.
See also A177405/A177407.
Sequence in context: A132463 A153901 A132844 * A324797 A049456 A117506
KEYWORD
nonn,nice,frac,tabf,changed
EXTENSIONS
More terms from Robert G. Wilson v, Apr 08 2004
Changed offset (=order of first row) to 1 by R. J. Mathar, Apr 26 2009
STATUS
approved