A018805 -id:A018805 - cp's OEIS frontend

A342841 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 10^n.

1, 841, 832693, 832046137, 831916552903, 831908477106883, 831907430687799769, 831907383078281024371, 831907373418800027750413, 831907372722449100147414487, 831907372589073124899487831735, 831907372581823023465031521920149, 831907372580768386561159867257319711

Offset: 0

Karl-Heinz Hofmann, Mar 24 2021

			For visualization, the set(x, y, z) is inscribed in a cube matrix.
"o" stands for a gcd = 1.
"." stands for a gcd > 1.
.
For n=1, the size of the cube matrix is 10 X 10 X 10:
.
                         / : : : : : : : : : :
                        /                               100 Sum (z = 1)
                z = 7 |/1 2 3 4 5 6 7 8 9 10             |
                    --+---------------------             75 Sum (z = 2)
                   1 /| o o o o o o o o o o    10        |
                   2/ | o o o o o o o o o o    10        91 Sum (z = 3)
                   /                           10        |
           z = 8 |/1 2 3 4 5 6 7 8 9 10        10       75 Sum (z = 4)
               --+---------------------        10      /
              1 /| o o o o o o o o o o    10   10     96 Sum (z = 5)
              2/ | o . o . o . o . o .     5    9    /
              /                           10   10   67 Sum (z = 6)
      z = 9 |/1 2 3 4 5 6 7 8 9 10         5   10  /
          --+---------------------        10   10 /
         1 /| o o o o o o o o o o    10    5   --/
         2/ | o o o o o o o o o o    10   10   99 Sum (z = 7)
         /                            7    5   /
z = 10 |/1 2 3 4 5 6 7 8 9 10        10   10  /
     --+---------------------        10    5 /
     1 | o o o o o o o o o o    10    7   --/
     2 | o . o . o . o . o .     5   10   75 Sum (z = 8)
     3 | o o o o o o o o o o    10   10   /
     4 | o . o . o . o . o .     5    7  /
     5 | o o o o . o o o o .     8   10 /
     6 | o . o . o . o . o .     5   --/
     7 | o o o o o o o o o o    10   91 Sum (z = 9)
     8 | o . o . o . o . o .     5   /
     9 | o o o o o o o o o o    10  /
    10 | o . o . . . o . o .     4 /
                                --/
                                72 Sum (z = 10)
                                /
                               |
                            ------
                              841 Cube Sum (z = 1..10)

Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Chai Wah Wu, Table of n, a(n) for n = 0..15
Karl-Heinz Hofmann, An animation of the cube with n = 1.

Cf. A342586 (for 10^n X 10^n), A018805, A002117 (zeta(3)), A071778.
Related counts of k-tuples:
pairs: A018805, A342632, A342586;
triples: A071778, A342935, A342841;
quadruples: A082540, A343527, A343193;
5-tuples: A343282;
6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

import math
for n in range (0, 10):
     counter = 0
     for x in range (1, pow(10, n)+1):
        for y in range(1, pow(10, n)+1):
            for z in range(1, pow(10, n)+1):
                if math.gcd(math.gcd(y, x),z) ==  1:
                    counter += 1
     print(n, counter)

Lim_{n->infinity} a(n)/10^(3*n) = 1/zeta(3) = 1/Apéry's constant.

a(n) = A071778(10^n).

a(5)-a(10) from Hugo Pfoertner, Mar 25 2021
a(11) from Hugo Pfoertner, Mar 26 2021
a(12) from Bruce Garner, Mar 29 2021

A342935 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 2^n.

Original entry on oeis.org

1, 7, 55, 439, 3433, 27541, 218773, 1749223, 13964245, 111725197, 893433661, 7147232467, 57169672861, 457364647435, 3658819119307, 29270432746633, 234161501271463, 1873293863661469, 14986321908515773, 119890565631185995, 959124025074311215, 7672992332048493361

Offset: 0

Karl-Heinz Hofmann, Mar 29 2021

nonn

			For n=3, the size of the division cube matrix is 8 X 8 X 8:
.
                            :   : : : : : : : :
.
                        z = 4 | 1 2 3 4 5 6 7 8
                        ------+----------------------
                          1  /| o o o o o o o o    8
                          2 / | o . o . o . o .    4      64 Sum (z = 1)
                          3/  | o o o o o o o o    8      /
                          /                 o .    4    48  Sum (z = 2)
                  z = 5 |/1 2 3 4 5 6 7 8     o    8    /
                  ------+----------------------    4  60  Sum (z = 3)
                    1  /| o o o o o o o o    8     8  /
                    2 / | o o o o o o o o    8     4 /
                    3/  | o o o o o o o o    8    --/
                    /                 o o    8    48  Sum (z = 4)
            z = 6 |/1 2 3 4 5 6 7 8     o    7    /
            ------+----------------------    8   /
              1  /| o o o o o o o o    8     8  /
              2 / | o . o . o . o .    4     8 /
              3/  | o o o o o o o o    6    --/
              /                 o .    4    63  Sum (z = 5)
      z = 7 |/1 2 3 4 5 6 7 8     o    8    /
      ------+----------------------    3   /
        1  /| o o o o o o o o    8     8  /
        2 / | o o o o o o o o    8     4 /
        3/  | o o o o o o o o    8    --/
        /                 o o    8    45  Sum (z = 6)
z = 8 |/1 2 3 4 5 6 7 8     o    8    /
------+----------------------    8   /
  1   | o o o o o o o o    8     7  /
  2   | o . o . o . o .    4     8 /
  3   | o o o o o o o o    8    --/
  4   | o . o . o . o .    4    63  Sum (z = 7)
  5   | o o o o o o o o    8    /
  6   | o . o . o . o .    4   /
  7   | o o o o o o o o    8  /
  8   | o . o . o . o .    4 /
                          --/
                          48  Sum (z = 8)
                           |
                         ---
                         439  Cube Sum (z = 1..8)

Chai Wah Wu, Table of n, a(n) for n = 0..53 (terms n = 0..32 from Karl-Heinz Hofmann)

Cf. A088453, A002117, A018805, A342632, A342586, A071778, A342841.

Array[Sum[MoebiusMu[k]*Floor[(2^#)/k]^3, {k, 2^# + 1}] &, 22, 0] (* Michael De Vlieger, Apr 05 2021 *)

from labmath import mobius
def A342935(n): return sum(mobius(k)*(2**n//k)**3 for k in range(1, 2**n+1))

Lim_{n->infinity} a(n)/2^(3*n) = 1/zeta(3) = A088453 = 1/Apéry's constant.

a(n) = A071778(2^n).

Edited by N. J. A. Sloane, Jun 13 2021

A046657 a(n) = A002088(n)/2.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 14, 16, 21, 23, 29, 32, 36, 40, 48, 51, 60, 64, 70, 75, 86, 90, 100, 106, 115, 121, 135, 139, 154, 162, 172, 180, 192, 198, 216, 225, 237, 245, 265, 271, 292, 302, 314, 325, 348, 356, 377, 387, 403, 415, 441, 450, 470, 482

Offset: 2

N. J. A. Sloane

nonn

a(n) = |{(x,y) : 1 <= x <= y <= n, x + y <= n, 1 = gcd(x,y)}| = |{(x,y) : 1 <= x <= y <= n, x + y > n, 1 = gcd(x,y)}|. - Steve Butler, Mar 31 2006

Brousseau proved that if the starting numbers of a generalized Fibonacci sequence are <= n (for n > 1) then the number of such sequences with relatively prime successive terms is a(n). - Amiram Eldar, Mar 31 2017

Giovanni Resta, Table of n, a(n) for n = 2..10000
Alfred Brousseau, A Note on the Number of Fibonacci Sequences, The Fibonacci Quarterly, Vol. 10, No. 6 (1972), pp. 657-658.

Cf. A002088.

Partial sums of A023022.

List([2..60],n->Sum([1..n],k->Phi(k)/2)); # Muniru A Asiru, Mar 05 2018

a:=n->sum(numtheory[phi](k),k=1..n): seq(a(n)/2, n=2..60); # Muniru A Asiru, Mar 05 2018

Rest@ Accumulate[EulerPhi@ Range@ 56]/2 (* Michael De Vlieger, Apr 02 2017 *)

a(n) = sum(k=1, n, eulerphi(k))/2; \\ Michel Marcus, Apr 01 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A046657(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(4*A046657(k1)-1)
        j, k1 = j2, n//j2
    return (n*(n-1)-c+j)//4 # Chai Wah Wu, Mar 25 2021

a(n) = 1/2 + Sum_{iJoseph Wheat, Feb 22 2018

A100613 Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y) > 1}.

Original entry on oeis.org

0, 1, 2, 5, 6, 13, 14, 21, 26, 37, 38, 53, 54, 69, 82, 97, 98, 121, 122, 145, 162, 185, 186, 217, 226, 253, 270, 301, 302, 345, 346, 377, 402, 437, 458, 505, 506, 545, 574, 621, 622, 681, 682, 729, 770, 817, 818, 881, 894, 953, 990, 1045, 1046, 1117, 1146, 1209

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 02 2004

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A018805, A000290, A185670, A063985, A242114, A229099.

a100613 n = length [()| x <- [1..n], y <- [1..n], gcd x y > 1]
-- Reinhard Zumkeller, Jan 21 2013

f[n_] := Table[ #^2 &[m], {m, 1, n + 1}] - FoldList[Plus, 1, 2 Array[EulerPhi, n, 2]] (* Gregg K. Whisler, Jun 25 2008 *)

a(n) = sum(i=1, n, sum(j=1, n, gcd(i,j)>1)); \\ Michel Marcus, Jan 30 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A100613(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(k1**2-A100613(k1))
        j, k1 = j2, n//j2
    return n+c-j # Chai Wah Wu, Mar 24 2021

a(n) = A000290(n) - A018805(n) = A185670(n) + A063985(n). - Reinhard Zumkeller, Jan 21 2013
a(n) = Sum_{k = 2..n} A242114(n,k). - Reinhard Zumkeller, May 04 2014
a(n) ~ kn^2, where k = 1 - 6/Pi^2 = 0.39207... (A229099). - Charles R Greathouse IV, Mar 29 2024

A213212 Number of distinct products ijk over all triples (i,j,k) with i,j,k >= 0 and i+j+k <= n and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 10, 12, 17, 20, 26, 29, 38, 44, 52, 59, 72, 78, 94, 104, 118, 130, 149, 160, 182, 198, 221, 237, 263, 278, 308, 330, 361, 383, 416, 438, 480, 509, 546, 574, 620, 646, 699, 734, 777, 816, 872, 907, 969, 1012, 1071, 1117, 1190, 1233, 1307, 1361

Offset: 0

Robert Price, Mar 02 2013

This sequence is in reply to an extension request made in A100450.

Note that gcd(0,m) = m for any m.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Robert Price)

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213213.

h:= proc() true end:
b:= proc(n) local c, i, j, p;
      c:=0;
      for i to iquo(n, 3) do
        for j from i to iquo(n-i, 2) do
          if igcd(i, j, n-i-j)=1 then p:= i*j*(n-i-j);
            if h(p) then h(p):= false; c:=c+1 fi
          fi
        od
      od; c
    end:
a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 02 2013

f[n_] := Length[ Union[ Flatten[ Table[ If[ i+j+k <= n&& GCD[i, j, k] <= 1, i*j*k, 0], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]]]; Table[ f[n], {n, 0, 200}]

a(n) = (A213208(n) + 1)/2.

A213213 Number of distinct products ijk over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 28, 33, 40, 46, 56, 64, 75, 84, 97, 109, 125, 137, 156, 170, 192, 210, 232, 251, 276, 296, 322, 347, 376, 400, 435, 463, 498, 529, 567, 600, 641, 674, 720, 758, 808, 849, 901, 942, 1001, 1051, 1110, 1157, 1225, 1275

Offset: 0

Robert Price, Mar 02 2013

This sequence is in reply to an extension request made in A100450.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Robert Price)
Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See p. 20.

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212.

h:= proc() true end:
b:= proc(n) local c, i, j, p;
      c:=0;
      for i to iquo(n, 3) do
        for j from i to iquo(n-i, 2) do p:= i*j*(n-i-j);
          if h(p) then h(p):= false; c:=c+1 fi
        od
      od; c
    end:
a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 02 2013

f[n_] := Length[ Union[ Flatten[ Table[ If[ i+j+k <= n, i*j*k, 0], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]]]; Table[ f[n], {n, 0, 200}]

a(n) = (A213207(n)+1)/2.

A333295 Triangle read by rows: T(m,n) = Sum_{1 <= i <= m, 1 <= j <= n, gcd{i,j}=1} 1, where m >= n >= 1.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 4, 6, 9, 11, 5, 8, 12, 15, 19, 6, 9, 13, 16, 21, 23, 7, 11, 16, 20, 26, 29, 35, 8, 12, 18, 22, 29, 32, 39, 43, 9, 14, 20, 25, 33, 36, 44, 49, 55, 10, 15, 22, 27, 35, 38, 47, 52, 59, 63, 11, 17, 25, 31, 40, 44, 54, 60, 68, 73, 83, 12, 18, 26, 32, 42, 46, 57, 63, 71, 76, 87, 91

Offset: 1

N. J. A. Sloane, Mar 24 2020

This is the same triangle as A331781, except the initial row and column of 0's is missing.

			Triangle begins:
1,
2, 3,
3, 5, 7,
4, 6, 9, 11,
5, 8, 12, 15, 19,
6, 9, 13, 16, 21, 23,
7, 11, 16, 20, 26, 29, 35,
8, 12, 18, 22, 29, 32, 39, 43,
9, 14, 20, 25, 33, 36, 44, 49, 55,
10, 15, 22, 27, 35, 38, 47, 52, 59, 63,
11, 17, 25, 31, 40, 44, 54, 60, 68, 73, 83,
12, 18, 26, 32, 42, 46, 57, 63, 71, 76, 87, 91,
...

Cf. A331781. Main diagonal is A018805.

V0 := proc(m,n) local a,i,j; 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;
for m from 1 to 14 do lprint([seq(V0(m,n),n=1..m)]); od:

A343193 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 10^n.

Original entry on oeis.org

1, 9279, 92434863, 923988964495, 9239427676877311, 92393887177379735327, 923938441006918271400831, 9239384074081430755652624559, 92393840333765561759423951663423, 923938402972369921481535120722882015

Offset: 0

Karl-Heinz Hofmann, Apr 07 2021

nonn

			(1,2,2,3) is counted, but (2,4,4,6) is not, because gcd = 2.
For n=1, the size of the division tesseract matrix is 10 X 10 X 10 X 10:
.
              o------------x(w=10)------------o
             /|.                            ./ |
            / |.                           ./  |
           /  |.                          ./   |
          /   |.                         ./    |
         /    |.                      z(w=10)  |
        /     |.                      . /      |
       /      |.                     . /       |
      /       |.                   .  /     y(w=10)
     o------------------------------.o         |
    |\        /|¯¯¯¯¯¯x(w=1)¯¯¯¯¯¯/. |         |
    | w      / |                 /.| |         |
    |  \ z(w=1)|                /. | |         |
    |   \  /   |y(w=1)         /.  | |         |
    |    \/-------------------/.   | |         |
    |     |                   |    | |         |        w | sums
    |     |  Cube at w = 1    |    | |         |      ----+-----
    |     |   10 X 10 X 10    | _ _| |---------o        1 | 1000
    |     |    contains       |    / |         /        2 |  875
    |     |      1000         |   /  |        /         3 |  973
    |     |    completely     |  /   |       /          4 |  875
    |     | reduced fractions | /    |      /           5 |  992
    |     |                   |/     |     /            6 |  849
    |     /------------------- \     |    /             7 |  999
    |    /                      \    |   /              8 |  875
    |   w                        w   |  /               9 |  973
    |  /                          \  | /               10 |  868
    | /                            \ |/               ----+-----
    o -------------------------------o       sum for a(1) | 9279

Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Chai Wah Wu, Table of n, a(n) for n = 0..15

Cf. A215267, A013662, A082540, A342841 (3D), A342586 (2D).
Related counts of k-tuples:
pairs: A018805, A342632, A342586;
triples: A071778, A342935, A342841;
quadruples: A082540, A343527, A343193;
5-tuples: A343282;
6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

from labmath import mobius
def A343193(n): return sum(mobius(k)*(10**n//k)**4 for k in range(1, 10**n+1))

Lim_{n->infinity} a(n)/10^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.

a(n) = A082540(10^n).

Edited by N. J. A. Sloane, Jun 13 2021

A343527 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 2^n.

Original entry on oeis.org

1, 15, 239, 3823, 60735, 972191, 15517679, 248252879, 3969108895, 63506982943, 1015951568815, 16255093526239, 260068569617727, 4161109496115135, 66577084386669199, 1065232436999055375, 17043668344393625999, 272698739815301095247, 4363176901343767529551, 69810828455823683068415, 1116973047989955380768527

Offset: 0

Karl-Heinz Hofmann, Apr 18 2021

nonn

			.
For n=3, the size of the gris is 8 X 8 X 8 X 8:
.
              o------------x(w=8)-------------o
             /|.                            ./ |
            / |.                           ./  |
           /  |.                          ./   |
          /   |.                         ./    |
         /    |.                      z(w=8)   |
        /     |.                      . /      |
       /      |.                     . /       |
      /       |.                   .  /     y(w=8)
     o------------------------------.o         |
    |\        /|¯¯¯¯¯¯x(w=1)¯¯¯¯¯¯/. |         |
    | w      / |                 /.| |         |
    |  \ z(w=1)|                /. | |         |
    |   \  /   |y(w=1)         /.  | |         |
    |    \/-------------------/.   | |         |
    |     |                   |    | |         |        w | sums
    |     |  Cube at w = 1    |    | |         |      ----+-----
    |     |    8 X 8 X 8      | _ _| |---------o        1 |  512
    |     |    contains       |    / |         /        2 |  448
    |     |       512         |   /  |        /         3 |  504
    |     |    completely     |  /   |       /          4 |  448
    |     | reduced fractions | /    |      /           5 |  511
    |     |                   |/     |     /            6 |  441
    |     /------------------- \     |    /             7 |  511
    |    /                      \    |   /              8 |  448
    |   w                        w   |  /             ----+-----
    |  /                          \  | /     sum for a(3) | 3823
    | /                            \ |/
    o -------------------------------o

Chai Wah Wu, Table of n, a(n) for n = 0..52 (n = 0..31 from Karl-Heinz Hofmann)

Cf. A018805, A342632, A342586, A071778.

Cf. A342935, A342841, A082540, A343193.

from labmath import mobius
def A343527(n): return sum(mobius(k)*(2**n//k)**4 for k in range(1, 2**n+1))

Lim_{n->infinity} a(n)/2^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.

a(n) = A082540(2^n).

Edited by N. J. A. Sloane, Jun 13 2021

A344527 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of ordered k-tuples (x_1, x_2, ..., x_k) with gcd(x_1, x_2, ..., x_k) = 1 (1 <= {x_1, x_2, ..., x_k} <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 11, 1, 1, 31, 79, 55, 19, 1, 1, 63, 241, 239, 115, 23, 1, 1, 127, 727, 991, 607, 181, 35, 1, 1, 255, 2185, 4031, 3091, 1199, 307, 43, 1, 1, 511, 6559, 16255, 15559, 7501, 2303, 439, 55, 1, 1, 1023, 19681, 65279, 77995, 45863, 16531, 3823, 637, 63, 1

Offset: 1

Seiichi Manyama, May 22 2021

			G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} mu(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  3,   7,   15,   31,    63, ...
  1,  7,  25,   79,  241,   727, ...
  1, 11,  55,  239,  991,  4031, ...
  1, 19, 115,  607, 3091, 15559, ...
  1, 23, 181, 1199, 7501, 45863, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened.

Columns k=1..6 give A000012, A018805, A071778, A082540, A082544, A343978.
T(n,n) gives A332468.
Cf. A343510, A343516, A344479.

T[n_, k_] := Sum[MoebiusMu[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)

T(n, k) = sum(j=1, n, moebius(j)*(n\j)^k);

PARI
```
T(n, k) = n^k-sum(j=2, n, T(n\j, k));
```

from functools import lru_cache
from itertools import count, islice
@lru_cache(maxsize=None)
def A344527_T(n,k):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A344527_T(k1,k)
        j, k1 = j2, n//j2
    return n*(n**(k-1)-1)-c+j
def A344527_gen(): # generator of terms
    return (A344527_T(k+1, n-k) for n in count(1) for k in range(n))
A344527_list = list(islice(A344527_gen(),30)) # Chai Wah Wu, Nov 02 2023

G.f. of column k: (1/(1 - x)) * Sum_{i>=1} mu(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
T(n,k) = Sum_{j=1..n} mu(j) * floor(n/j)^k.
T(n,k) = n^k - Sum_{j=2..n} T(floor(n/j),k).

cp's OEIS Frontend

A342841 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 10^n.

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A342935 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 2^n.

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A046657 a(n) = A002088(n)/2.

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A100613 Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y) > 1}.

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A213212 Number of distinct products i*j*k over all triples (i,j,k) with i,j,k >= 0 and i+j+k <= n and gcd(i,j,k) <= 1.

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A213213 Number of distinct products i*j*k over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.

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A333295 Triangle read by rows: T(m,n) = Sum_{1 <= i <= m, 1 <= j <= n, gcd{i,j}=1} 1, where m >= n >= 1.

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A213212 Number of distinct products ijk over all triples (i,j,k) with i,j,k >= 0 and i+j+k <= n and gcd(i,j,k) <= 1.

A213213 Number of distinct products ijk over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.