A049691 a(n)=T(n,n), array T as in A049687. Also a(n)=T(2n,2n), array T given by A049639.

0, 3, 5, 9, 13, 21, 25, 37, 45, 57, 65, 85, 93, 117, 129, 145, 161, 193, 205, 241, 257, 281, 301, 345, 361, 401, 425, 461, 485, 541, 557, 617, 649, 689, 721, 769, 793, 865, 901, 949, 981, 1061, 1085, 1169, 1209, 1257, 1301, 1393, 1425, 1509, 1549

Offset

0,2

Comments

a(n) is related to the sequence b(n) = |{(x, y): gcd(x, y) = 1, 1<=x, y<=n}| (A018805) as follows: a(n) = b(n - 1) + 2 (for n > 1). - Shawn Westmoreland (westmore(AT)math.utexas.edu), Jun 11 2003

Comment from N. J. A. Sloane, Sep 08 2019 (Start)

The above comment can be rephrased as saying that a(n) is the cardinality of the subsequence F(B(2n),n) of the Farey series F_{2n} that is extensively studied in Matveev (2017). See the definition on page 1.

For example, F(B(2),1), F(B(4),2), F(B(6),3), and F(B(8),4) are:

[0, 1/2, 1],

[0, 1/3, 1/2, 2/3, 1],

[0, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1],

[0, 1/5, 1/4, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 3/4, 4/5, 1],

of cardinalities 3,5,9,13 respectively. See also A324796/A324797. (End)

a(n) is the number of visible points on an n X n square lattice when viewed from (0, 0), (0, n), (n, 0), or (n, n). - Torlach Rush, Nov 16 2020

Also number of elements in { c/d ; -d <= c <= d <= n }, i.e., distinct fractions with denominator not exceeding n and absolute value of numerator not exceeding the denominator. - M. F. Hasler, Mar 26 2023

References

A. O. Matveev, Farey Sequences, De Gruyter, 2017.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A. O. Matveev, Farey Sequences: Errata + Haskell code

Eric Weisstein's World of Mathematics, Visible Point

Formula

a(n) = A206297(n+1) = 2 + A018805(n) for n > 0. - Andrew Howroyd, Sep 17 2017

a(n) = 1 + 2 * Sum{k=1..n} A000010(k), n > 0. - Torlach Rush, Nov 24 2020

Maple

Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end: # A006842/A006843
BF := proc(m) local a, i, h, k; global Farey; a:=[];
for i in Farey(2*m) do h:=numer(i); k:=denom(i);
if (h <= m) and (k-m <= h) then a:=[op(a), i]; fi; od: a; end;
[seq(nops(BF(m), m=1..20)]; # this sequence - N. J. A. Sloane, Sep 08 2019

Mathematica

a[0] = 0; a[n_] := 2 + Sum[Quotient[n, g]^2*MoebiusMu[g], {g, 1, n}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 07 2017, translated from PARI *)

Prog

(PARI) a(n) = if(n>0, 2, 0) + sum(g=1, n, (n\g)^2 * moebius(g)); \\ Andrew Howroyd, Sep 17 2017
(PARI) a(n) = if(n>0, 1, 0) + 2 * sum(k=1, n, eulerphi(k)); \\ Torlach Rush, Nov 24 2020
(PARI) a(n)=#Set(concat([[c/d|c<-[-d..d], d]|d<-[0..n]])) \\ For illustrative purpose only! - M. F. Hasler, Mar 26 2023
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A049691(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(A049691(k1)-2)
        j, k1 = j2, n//j2
    return n*(n-1)-c+j+2 # Chai Wah Wu, Aug 04 2024

Crossrefs

Cf. A000010, A018805, A049687, A324796, A324797.

A206297 is an essentially identical sequence.

Sequence in context: A249424 A076274 A058989 * A206297 A320596 A227565

Adjacent sequences: A049688 A049689 A049690 * A049692 A049693 A049694

Keyword

nonn

Author

Clark Kimberling

Extensions

Terms a(41) and beyond from Andrew Howroyd, Sep 17 2017

Status

approved