A018805 -id:A018805 - cp's OEIS frontend

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

Clark Kimberling

nonn

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

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

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

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

A206297 is an essentially identical sequence.

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

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 *)

a(n) = if(n>0, 2, 0) + sum(g=1, n, (n\g)^2 * moebius(g)); \\ Andrew Howroyd, Sep 17 2017

a(n) = if(n>0, 1, 0) + 2 * sum(k=1, n, eulerphi(k)); \\ Torlach Rush, Nov 24 2020

a(n)=#Set(concat([[c/d|c<-[-d..d],d]|d<-[0..n]])) \\ For illustrative purpose only! - M. F. Hasler, Mar 26 2023

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

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

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

A071778 Number of ordered triples (a, b, c) with gcd(a, b, c) = 1 and 1 <= {a, b, c} <= n.

Original entry on oeis.org

1, 7, 25, 55, 115, 181, 307, 439, 637, 841, 1171, 1447, 1915, 2329, 2881, 3433, 4249, 4879, 5905, 6745, 7861, 8911, 10429, 11557, 13297, 14773, 16663, 18355, 20791, 22495, 25285, 27541, 30361, 32905, 36289, 38845, 42841, 46027, 49987, 53395

Offset: 1

Michael Malak (mmalak(AT)alum.mit.edu), Jun 04 2002

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
IBM Ponder This, Coin-weighing problem, Jun 01 2002
Eric Weisstein's World of Mathematics, Greatest Common Divisor

Cf. A018805 (ordered pairs), A082540, A082544, A343978, A344522.

public class Triples { public static void main(String[] argv) { int i, j, k, a, m, n, d; boolean cf; try {a = Integer.parseInt(argv[0]);} catch (Exception e) {a = 10;}
for (m = 1; m <= a; m++) { n = 0; for (i = 1; i <= m; i++) for (j = 1; j <= m; j++) for (k = 1; k <= m; k++) { cf = false; for (d = 2; d <= m; d++) cf = cf || ((i % d == 0) && (j % d == 0) && (k % d == 0)); if (!cf) n++; } System.out.println(m + ": " + n); } } }

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from 1 to n do for j from 1 to n do t2:=gcd(i,j); for k from 1 to n do t3:=gcd(t2,k); if t3 = 1 then t1:=t1+1; fi; od: od: od: t1; end;

a[n_] := Sum[MoebiusMu[k]*Quotient[n, k]^3, {k, 1, n}]; Array[a, 40] (* Jean-François Alcover, Apr 14 2014, after Benoit Cloitre *)

PARI
```
a(n)=sum(k=1,n,moebius(k)*(n\k)^3)
```

a(n)=my(s); forsquarefree(k=1,n, s+=moebius(k)*(n\k[1])^3); s \\ Charles R Greathouse IV, Jan 08 2018

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3)/(1-x)) \\ Seiichi Manyama, May 22 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A071778(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A071778(k1)
        j, k1 = j2, n//j2
    return n*(n**2-1)-c+j # Chai Wah Wu, Mar 29 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^3. - Benoit Cloitre, May 11 2003
a(n) = n^3 - Sum_{j=2..n} a(floor(n/j)). - Vladeta Jovovic, Nov 30 2004
G.f.: (1/(1 - x)) * Sum_{k >= 1} mu(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3. - Seiichi Manyama, May 22 2021
a(n) ~ n^3/zeta(3). - Vaclav Kotesovec, Sep 14 2021

A358298 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of lines defining the Farey diagram Farey(n,k) of order (n,k).

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 6, 11, 11, 6, 8, 19, 20, 19, 8, 12, 29, 36, 36, 29, 12, 14, 43, 52, 60, 52, 43, 14, 20, 57, 78, 88, 88, 78, 57, 20, 24, 77, 100, 128, 124, 128, 100, 77, 24, 30, 97, 136, 162, 180, 180, 162, 136, 97, 30, 34, 121, 166, 216, 224, 252, 224, 216, 166, 121, 34

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

We work with lines with equation ux + vy + w = 0 in the (x,y) plane.
This line has slope -u/v, and crosses the vertical y axis at the intercept point y = -w/v
For the Farey diagram Farey(m,n), u is an integer between -(m-1) and +(m-1), v is between -(n-1) and +(n-1) and w can be any integer.
The only lines that are used are those that hit the unit square 0 <= x <= 1, 0 <= y <= 1 in at least two points.
This means that we only need to look at w's with |w| <=  |u| + |v|.
T(m,n) is the number of such lines.
For illustrations of Farey(3,3) and Farey(3,4) see Khoshnoudirad (2015), Fig. 2, and Darat et al. (2009), Fig. 2. For further illustrations see A358882-A358885.

			The full array T(n,k), n >= 0, k>= 0, begins:
  2, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60,  ...
  3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205,  ...
  4, 11, 20, 36, 52, 78, 100, 136, 166, 210, 246, 302,  ...
  6, 19, 36, 60, 88, 128, 162, 216, 266, 326, 386, 468, ...
  8, 29, 52, 88, 124, 180, 224, 298, 360, 444, 518, 628, ...
  12, 43, 78, 128, 180, 252, 316, 412, 498, 608, 706,  ...
  14, 57, 100, 162, 224, 316, 388, 508, 608, 738, 852, ...
  ...

Alain Daurat, M. Tajine, and M. Zouaoui, About the frequencies of some patterns in digital planes. Application to area estimators. Computers & Graphics. 33.1 (2009), 11-20.
Daniel Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures, Applicable Analysis and Discrete Mathematics, 9 (2015), 73-84; doi:10.2298/AADM150219008K. See Theorem 1, |DF(m,n)|.

Cf. A358299.
Row 0 is essentially A225531, row 1 is A358300, main diagonal is A358301.
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
Amn:=proc(m,n) local a,i,j;  # A331781 or equally A333295. Diagonal is A018805.
a:=0; for i from 1 to m do for j from 1 to n do
if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
# The present sequence is:
Dmn:=proc(m,n) local d,t1,u,v,a; global A005728, Amn;
a:=A005728(m)+A005728(n);
t1:=0; for u from 1 to m do for v from 1 to n do
d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
a+2*t1-2*Amn(m,n); end;
for m from 1 to 8 do lprint([seq(Dmn(m,n),n=1..20)]); od:

A005728[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
Amn[m_, n_] := Module[{a, i, j}, a = 0; For[i = 1, i <= m, i++, For[j = 1, j <= n, j++, If[GCD[i, j] == 1, a = a + 1]]]; a];
Dmn[m_, n_] := Module[{d, t1, u, v, a}, a = A005728[m] + A005728[n]; t1 = 0; For[u = 1, u <= m, u++, For[v = 1, v <= n, v++, d = GCD[u, v]; If[d >= 1 , t1 = t1 + (u + v)* EulerPhi[d]/d]]]; a + 2*t1 - 2*Amn[m, n]];
Table[Dmn[m - n, n], {m, 0, 10}, {n, 0, m}] // Flatten (* Jean-François Alcover, Apr 03 2023, after Maple code *)

A100450 Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 7, 19, 51, 99, 195, 291, 483, 675, 963, 1251, 1731, 2115, 2787, 3363, 4131, 4899, 6051, 6915, 8355, 9507, 11043, 12483, 14595, 16131, 18531, 20547, 23139, 25443, 28803, 31107, 34947, 38019, 41859, 45315, 49923, 53379, 58851, 63171, 68547

Offset: 0

N. J. A. Sloane, Nov 21 2004

Note that gcd(0,m) = m for any m.
I would also like to get the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with |i| + |j| + |k| <= n; also over all ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.
Also the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0 and i + j + k = n; also over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.
Also the number of ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.
From Robert Price, Mar 05 2013: (Start)
The sequences that address the previous comments are:
Distinct sums i+j+k with or without the GCD qualifier results in a(n)=2n+1 (A005408).
Distinct products i*j*k without the GCD qualifier is given by A213207.
Distinct products i*j*k with    the GCD qualifier is given by A213208.
With the restriction i,j,k >= 0 ...
  Distinct sums or products equal to n is trivial and always equals one (A000012).
  Distinct sums <= n results in a(n)=n (A001477).
  Distinct products <= n without the GCD qualifier is given by A213213.
  Distinct products <= n with    the GCD qualifier is given by A213212.
  Ordered triples with sum = n  without the GCD qualifier is A000217(n+1).
  Ordered triples with sum = n  with    the GCD qualifier is A048240.
  Ordered triples with sum <= n without the GCD qualifier is A000292.
  Ordered triples with sum <= n with    the GCD qualifier is A048241. (End)
This sequence (A100450) without the GCD qualifier results in A001845. - Robert Price, Jun 04 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000124, A000292, A018805, A027430, A048240, A048241, A100448, A100449, A213207, A213208, A213212, A213213.

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from -n to n do for j from -n to n do t2:=gcd(i,j); for k from -n to n do if abs(i) + abs(j) + abs(k) <= n then t3:=gcd(t2,k); if t3 <= 1 then t1:=t1+1; fi; fi; od: od: od: t1; end;

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] + Abs[k] <= n && GCD[i, j, k] <= 1, {i, j, k}, {0, 0, 0}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]]]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v, Dec 14 2004 *)

G.f.: (3 + Sum_{k>=1} (moebius(k)*((1+x^k)/(1-x^k))^3))/(1-x). - Vladeta Jovovic, Nov 22 2004. [Sketch of proof: Let b(n) = number of ordered triples (i, j, k) with |i| + |j| + |k| = n and gcd(i, j, k) <= 1. Then a(n) = A100450(n) = partial sums of b(n) and Sum_{d divides n} b(d) = 4*n^2+2 = A005899(n) with g.f. ((1+x)/(1-x))^3.]

A082540 Number of ordered quadruples (a,b,c,d) with gcd(a,b,c,d)=1 (1 <= {a,b,c,d} <= n).

Original entry on oeis.org

1, 15, 79, 239, 607, 1199, 2303, 3823, 6223, 9279, 13919, 19183, 27007, 35743, 47519, 60735, 78719, 97103, 122447, 148527, 181839, 216959, 262543, 306863, 365343, 423855, 495855, 569055, 661679, 748527, 862047, 972191, 1104831, 1237247

Offset: 1

Benoit Cloitre, May 11 2003

nonn

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Column k=4 of A344527.
Cf. A018805, A071778, A082544, A343978.
Cf. A015634.

a(n)=sum(k=1,n,moebius(k)*floor(n/k)^4)

from functools import lru_cache
@lru_cache(maxsize=None)
def A082540(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082540(k1)
        j, k1 = j2, n//j2
    return n*(n**3-1)-c+j # Chai Wah Wu, Mar 29 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^4.
a(n) is asymptotic to c*n^4 with c=0.92393....
Lim_{n->infinity} a(n)/n^4 = 1/zeta(4) = A215267 = 90/Pi^4. - Karl-Heinz Hofmann, Apr 11 2021
Lim_{n->infinity} n^4/a(n) =   zeta(4) = A013662 = Pi^4/90. - Karl-Heinz Hofmann, Apr 11 2021
a(n) = n^4 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024

A100448 Number of triples (i,j,k) with 1 <= i <= j < k <= n and gcd{i,j,k} = 1.

Original entry on oeis.org

0, 1, 4, 9, 19, 30, 51, 73, 106, 140, 195, 241, 319, 388, 480, 572, 708, 813, 984, 1124, 1310, 1485, 1738, 1926, 2216, 2462, 2777, 3059, 3465, 3749, 4214, 4590, 5060, 5484, 6048, 6474, 7140, 7671, 8331, 8899, 9719, 10289, 11192, 11902, 12754, 13535, 14616

Offset: 1

N. J. A. Sloane, Nov 21 2004

Probably the partial sums of A102309. - Ralf Stephan, Jan 03 2005

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A015616, A015631, A027430, A071778, A018805.

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from 1 to n do for j from i to n do t2:=gcd(i,j); for k from j+1 to n do t3:=gcd(t2,k); if t3 = 1 then t1:=t1+1; fi; od: od: od: t1; end;

f[n_] := Length[ Union[ Flatten[ Table[ If[ GCD[i, j, k] == 1, {i, j, k}], {i, n}, {j, i, n}, {k, j + 1, n}], 2]]]; Table[ If[n > 3, f[n] - 1, f[n]], {n, 47}] (* Robert G. Wilson v, Dec 14 2004 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A100448(n):
    if n == 0:
        return 0
    c, j = 2, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(6*A100448(k1)+1)
        j, k1 = j2, n//j2
    return (n*(n**2-1)-c+j)//6 # Chai Wah Wu, Mar 29 2021

a(n) = (A071778(n)-1)/6. - Vladeta Jovovic, Nov 30 2004

a(n) = (1/6)*(-1 + Sum_{k=1..n} moebius(k)*floor(n/k)^3). - Ralf Stephan, Jan 03 2005

More terms from Robert G. Wilson v, Dec 14 2004

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A100449 Number of ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.

Original entry on oeis.org

1, 5, 9, 17, 25, 41, 49, 73, 89, 113, 129, 169, 185, 233, 257, 289, 321, 385, 409, 481, 513, 561, 601, 689, 721, 801, 849, 921, 969, 1081, 1113, 1233, 1297, 1377, 1441, 1537, 1585, 1729, 1801, 1897, 1961, 2121, 2169, 2337, 2417, 2513, 2601, 2785, 2849, 3017

Offset: 0

N. J. A. Sloane, Nov 21 2004

nonn

Note that gcd(0,m) = m for any m.
I would also like to get the sequences of the numbers of distinct sums i+j (also distinct products i*j) over all ordered pairs (i,j) with |i| + |j| <= n; also over all ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.
From Robert Price, May 10 2013: (Start)
List of sequences that address these extensions:
Distinct sums i+j with or without the GCD qualifier results in a(n)=2n+1 (A005408).
Distinct products i*j without the GCD qualifier is given by A225523.
Distinct products i*j with    the GCD qualifier is given by A225526.
With the restriction i,j >= 0 ...
Distinct sums or products equal to n is trivial and always equals one (A000012).
Distinct sums <=n with or without the GCD qualifier results in a(n)=n (A001477).
Distinct products <=n without the GCD qualifier is given by A225527.
Distinct products <=n with    the GCD qualifier is given by A225529.
Ordered pairs with the sum = n without the GCD qualifier is a(n)=n+1.
Ordered pairs with the sum = n with    the GCD qualifier is A225530.
Ordered pairs with the sum <=n without the GCD qualifier is A000217(n+1).
Ordered pairs with the sum <=n with    the GCD qualifier is A225531.
(End)
This sequence (A100449) without the GCD qualifier results in A001844. - Robert Price, Jun 04 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A018805, A100448, A100450, A027430, etc.

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from -n to n do for j from -n to n do if abs(i) + abs(j) <= n then t2:=gcd(i,j); if t2 <= 1 then t1:=t1+1; fi; fi; od: od: t1; end;
# second Maple program:
b:= proc(n) b(n):= numtheory[phi](n)+`if`(n=0, 0, b(n-1)) end:
a:= n-> 1+4*b(n):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 01 2013

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] <= n && GCD[i, j] <= 1, {i, j}, {0, 0}], {i, -n, n}, {j, -n, n}], 1]]]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 14 2004 *)

a(n) = 1+4*sum(k=1, n, eulerphi(k) ); \\ Joerg Arndt, May 10 2013

from functools import lru_cache
@lru_cache(maxsize=None)
def A100449(n):
    if n == 0:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*((A100449(k1)-3)//2)
        j, k1 = j2, n//j2
    return 2*(n*(n-1)-c+j)+1 # Chai Wah Wu, Mar 29 2021

a(n) = 1 + 4*Sum(phi(k), k=1..n) = 1 + 4*A002088(n). - Vladeta Jovovic, Nov 25 2004

More terms from Vladeta Jovovic, Nov 25 2004

A140434 Number of new visible points created at each step in an n X n grid.

Original entry on oeis.org

1, 2, 4, 4, 8, 4, 12, 8, 12, 8, 20, 8, 24, 12, 16, 16, 32, 12, 36, 16, 24, 20, 44, 16, 40, 24, 36, 24, 56, 16, 60, 32, 40, 32, 48, 24, 72, 36, 48, 32, 80, 24, 84, 40, 48, 44, 92, 32, 84, 40, 64, 48, 104, 36, 80, 48, 72, 56, 116, 32, 120

Offset: 1

Gregg Whisler, Jun 25 2008, Jun 28 2008

nonn

Equals row sums of triangle A143467. - Gary W. Adamson, Aug 17 2008
Equals first differences of A018805: (1, 3, 7, 11, 19, 23, 35, ...). - Gary W. Adamson, Aug 17 2008
a(n) is the number of rationals p/q such that |p| + |q| = n. - Geoffrey Critzer, Oct 11 2011
a(n) is the number of nonempty lists of positive integers whose continuants are equal to n. For example, for n = 6 these continuants are [6], [5,1], [1,5], and [1,4,1]. - Jeffrey Shallit, May 18 2016
a(n) is the number of Christoffel words of length n, for n>=2. Here a binary word w is a Christoffel word if its first and last letters are different, say w = axb with a<>b, and x is a palindrome, and w is the concatenation of two palindromes.   See the book of Reutenauer. - Jeffrey Shallit, Apr 04 2024

			G.f. = x + 2*x^2 + 4*x^3 + 4*x^4 + 8*x^5 + 4*x^6 + 12*x^7 + 8*x^8 + 12*x^9 + ...

C. Reutenauer, From Christoffel words to Markoff numbers, Oxford University Press, 2019.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A018805, A100613, A140435. Equals twice A000010 (for n >= 2).

a140434 n = a140434_list !! (n-1)
a140434_list = 1 : zipWith (-) (tail a018805_list) a018805_list
-- Reinhard Zumkeller, May 04 2014

f[n_] := FoldList[Plus, 1, 2 Array[EulerPhi, n, 2]] // Differences // Prepend[#, 1]&
a[ n_] := If[ n < 3, Max[0, n], Sum[ MoebiusMu[d] (2 n/d - 1 - Mod[n/d, 2]), {d, Divisors@n}]]; (* Michael Somos, Jul 24 2015 *)

{a(n) = if( n<3, max(0, n), sumdiv(n, d, moebius(d) * (2*n/d - 1 - (n/d)%2)))}; /* Michael Somos, Jul 24 2015 */

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+x^k)/(1-x^k)^2)) \\ Seiichi Manyama, May 24 2021

from sympy import totient
def A140434(n): return totient(n)<<1 if n>1 else 1 # Chai Wah Wu, May 09 2025

a(n) = 2*phi(n), where phi is Euler's phi function, A000010, for n >= 2.
Sum_{k=1..n} a(k)*floor(n/k) = n^2. - Benoit Cloitre, Nov 09 2016
G.f.: Sum_{k>=1} mu(k) * x^k * (1 + x^k)/(1 - x^k)^2. - Seiichi Manyama, May 24 2021

Mathematica simplified by Jean-François Alcover, Jun 06 2013

A331781 Triangle read by rows: T(m,n) = Sum_{0= n >= 1.

Original entry on oeis.org

0, 0, 1, 0, 2, 3, 0, 3, 5, 7, 0, 4, 6, 9, 11, 0, 5, 8, 12, 15, 19, 0, 6, 9, 13, 16, 21, 23, 0, 7, 11, 16, 20, 26, 29, 35, 0, 8, 12, 18, 22, 29, 32, 39, 43, 0, 9, 14, 20, 25, 33, 36, 44, 49, 55, 0, 10, 15, 22, 27, 35, 38, 47, 52, 59, 63, 0, 11, 17, 25, 31, 40, 44, 54, 60, 68, 73, 83

Offset: 1

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
0,
0, 1,
0, 2, 3,
0, 3, 5, 7,
0, 4, 6, 9, 11,
0, 5, 8, 12, 15, 19,
0, 6, 9, 13, 16, 21, 23,
0, 7, 11, 16, 20, 26, 29, 35,
0, 8, 12, 18, 22, 29, 32, 39, 43,
0, 9, 14, 20, 25, 33, 36, 44, 49, 55
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Lemma 11.

Main diagonal is A018805.

A333295 is essentially the same array.

VS := proc(m,n) local a,i,j; a:=0;
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
for m from 1 to 12 do lprint([seq(VS(m,n),n=1..m)]); od:

Table[Sum[Boole[# == 1] # &@ GCD[i, j], {i, m - 1}, {j, n - 1}], {m, 12}, {n, m}] // Flatten (* Michael De Vlieger, Feb 12 2020 *)

A015631 Number of ordered triples of integers from [ 1..n ] with no global factor.

Original entry on oeis.org

1, 3, 8, 15, 29, 42, 69, 95, 134, 172, 237, 287, 377, 452, 552, 652, 804, 915, 1104, 1252, 1450, 1635, 1910, 2106, 2416, 2674, 3007, 3301, 3735, 4027, 4522, 4914, 5404, 5844, 6432, 6870, 7572, 8121, 8805, 9389, 10249, 10831, 11776, 12506

Offset: 1

Olivier Gérard

nonn

Number of integer-sided triangles with at least two sides <= n and sides relatively prime. - Henry Bottomley, Sep 29 2006

			a(4) = 15 because the 15 triples in question are in lexicographic order: [1,1,1], [1,1,2], [1,1,3], [1,1,4], [1,2,2], [1,2,3], [1,2,4], [1,3,3], [1,3,4], [1,4,4], [2,2,3], [2,3,3], [2,3,4], [3,3,4] and [3,4,4]. - _Wolfdieter Lang_, Apr 04 2013
The a(4) = 15 triangles with at least two sides <= 4 and sides relatively prime (see _Henry Bottomley_'s comment above) are: [1,1,1], [1,2,2], [2,2,3], [1,3,3], [2,3,3], [2,3,4], [3,3,4], [3,3,5], [1,4,4], [2,4,5], [3,4,4], [3,4,5], [3,4,6], [4,4,5], [4,4,7]. - _Alois P. Heinz_, Feb 14 2020

R. J. Mathar, Table of n, a(n) for n = 1..10000

Column k=3 of A177976.

Cf. A000292, A002088, A015616, A015634, A015650, A134543.

[n eq 1 select 1 else Self(n-1)+ &+[MoebiusMu(n div d) *d*(d+1)/2:d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Feb 14 2020

with(numtheory):
b:= proc(n) option remember;
       add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
    end:
a:= proc(n) option remember;
      b(n) + `if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2011

a[1] = 1; a[n_] := a[n] = Sum[MoebiusMu[n/d]*d*(d+1)/2, {d, Divisors[n]}] + a[n-1]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jan 20 2014, after Maple *)
Accumulate[Table[Sum[MoebiusMu[n/d]*d*(d + 1)/2, {d, Divisors[n]}], {n, 1, 50}]] (* Vaclav Kotesovec, Jan 31 2019 *)

a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+1, 2))); \\ Seiichi Manyama, Jun 12 2021

a(n) = binomial(n+2, 3)-sum(k=2, n, a(n\k)); \\ Seiichi Manyama, Jun 12 2021

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^3)/(1-x)) \\ Seiichi Manyama, Jun 12 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A015631(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A015631(k1)
        j, k1 = j2, n//j2
    return n*(n-1)*(n+4)//6-c+j # Chai Wah Wu, Mar 30 2021

a(n) = (A071778(n)+3*A018805(n)+2)/6. - Vladeta Jovovic, Dec 01 2004
Partial sums of the Moebius transform of the triangular numbers (A007438). - Steve Butler, Apr 18 2006
a(n) = 2*A123324(n) - A046657(n) for n>1. - Henry Bottomley, Sep 29 2006
Row sums of triangle A134543. - Gary W. Adamson, Oct 31 2007
a(n) ~ n^3 / (6*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^3. - Ilya Gutkovskiy, Feb 14 2020
a(n) = n*(n+1)*(n+2)/6 - Sum_{j=2..n} a(floor(n/j)) = A000292(n) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

cp's OEIS Frontend

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

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A071778 Number of ordered triples (a, b, c) with gcd(a, b, c) = 1 and 1 <= {a, b, c} <= n.

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A358298 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of lines defining the Farey diagram Farey(n,k) of order (n,k).

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A100450 Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

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A082540 Number of ordered quadruples (a,b,c,d) with gcd(a,b,c,d)=1 (1 <= {a,b,c,d} <= n).

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A100448 Number of triples (i,j,k) with 1 <= i <= j < k <= n and gcd{i,j,k} = 1.

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A100449 Number of ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.

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