A342632 -id:A342632 - cp's OEIS frontend

A343978 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= n).

1, 63, 727, 4031, 15559, 45863, 116855, 257983, 526615, 983583, 1755143, 2935231, 4776055, 7407727, 11256623, 16498719, 23859071, 33434063, 46467719, 62949975, 84644439, 111486599, 146142583, 187854119, 240880239, 303814503, 382049919, 473813703, 586746719

Offset: 1

Karl-Heinz Hofmann, May 06 2021

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

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Joachim von zur Gathen and Jürgen Gerhard, Extract from "3.4. (Non-)Uniqueness of the gcd" chapter, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Column k=6 of A344527.
Cf. A343359, A013664, A018805, A071778, A082540, A082544.
Cf. A342632, A342586, A342935, A342841, A343527, A343193.

a(n)={sum(k=1, n+1, moebius(k)*(n\k)^6)} \\ Andrew Howroyd, May 08 2021

from labmath import mobius
def A343978(n): return sum(mobius(k)*(n//k)**6 for k in range(1, n+1))

from functools import lru_cache
@lru_cache(maxsize=None)
def A343978(n):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A343978(k1)
        j, k1 = j2, n//j2
    return n*(n**5-1)-c+j # Chai Wah Wu, May 17 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^6.
Lim_{n->infinity} a(n)/n^6 = 1/zeta(6) = A343359 = 945/Pi^6.
a(n) = n^6 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024

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

A342586 a(n) is the number of pairs (x,y) with 1 <= x, y <= 10^n and gcd(x,y)=1.

Original entry on oeis.org

1, 63, 6087, 608383, 60794971, 6079301507, 607927104783, 60792712854483, 6079271032731815, 607927102346016827, 60792710185772432731, 6079271018566772422279, 607927101854119608051819, 60792710185405797839054887, 6079271018540289787820715707, 607927101854027018957417670303

Offset: 0

Karl-Heinz Hofmann, Mar 16 2021

nonn

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

Joachim von zur Gathen and Jürgen Gerhard, Extract from "3.4. (Non-)Uniqueness of the gcd" chapter, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.
Hugo Pfoertner, Illustration of a(2)=6087.

a(n) = 2*A064018(n) - 1. - Hugo Pfoertner, Mar 16 2021
a(n) = A018805(10^n). - Michel Marcus, Mar 16 2021
Cf. A000010, A059956 (6/Pi^2), A342632, A342841, A343193, A343282.
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

a342586(n)=my(s, m=10^n); forfactored(k=1,m,s+=eulerphi(k)); s*2-1 \\ Bruce Garner, Mar 29 2021

a342586(n)=my(s, m=10^n); forsquarefree(k=1,m,s+=moebius(k)*(m\k[1])^2); s \\ Bruce Garner, Mar 29 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):
            if math.gcd(y,x) ==  1:
                counter += 1
     print(n, counter)

from functools import lru_cache
@lru_cache(maxsize=None)
def A018805(n):
  if n == 1: return 1
  return n*n - sum(A018805(n//j) for j in range(2, n//2+1)) - (n+1)//2
print([A018805(10**n) for n in range(8)]) # Michael S. Branicky, Mar 18 2021

Lim_{n->infinity} a(n)/10^(2*n) = 6/Pi^2 = 1/zeta(2).

a(10) from Michael S. Branicky, Mar 18 2021
More terms using A064018 from Hugo Pfoertner, Mar 18 2021
Edited by N. J. A. Sloane, Jun 13 2021

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

Original entry on oeis.org

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

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

A343282 Number of ordered 5-tuples (v,w, x, y, z) with gcd(v, w, x, y, z) = 1 and 1 <= {v, w, x, y, z} <= 10^n.

Original entry on oeis.org

1, 96601, 9645718621, 964407482028001, 96438925911789115351, 9643875373658964992585011, 964387358678775616636890654841, 96438734235127451288511508421855851, 9643873406165059293451290072800801506621

Offset: 0

Karl-Heinz Hofmann, Apr 10 2021

nonn

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. A082544, A013663, A342586, A342841, A343193.
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 A343282(n): return sum(mobius(k)*(10**n//k)**5 for k in range(1, 10**n+1))

Lim_{n->infinity} a(n)/10^(5*n) = 1/zeta(5) = A343308.

a(n) = A082544(10^n). - Chai Wah Wu, Apr 11 2021

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

A344038 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= 10^n).

Original entry on oeis.org

1, 983583, 983029267047, 982960635742968103, 982953384128772770413831, 982952672223441253533233827367, 982952600027678075050509511271466303, 982952593055042000417993486008754893529583, 982952592342881094406730790044111038427637071855

Offset: 0

Karl-Heinz Hofmann, May 07 2021

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

Cf. A342586, A342841, A343193, A343282, A343359, A013664.
Cf. A343978, A071778, A018805, A082540, A082544.
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

a(n)={sum(k=1, 10^n+1, moebius(k)*(10^n\k)^6)} \\ Andrew Howroyd, May 08 2021

from labmath import mobius
def A344038(n): return sum(mobius(k)*(10**n//k)**6 for k in range(1, 10**n+1))

Lim_{n->infinity} a(n)/10^(6*n) = 1/zeta(6) = A343359 = 945/Pi^4.

a(n) = A343978(10^n).

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

cp's OEIS Frontend

A343978 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= n).

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A342586 a(n) is the number of pairs (x,y) with 1 <= x, y <= 10^n and gcd(x,y)=1.

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

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

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A343282 Number of ordered 5-tuples (v,w, x, y, z) with gcd(v, w, x, y, z) = 1 and 1 <= {v, w, x, y, z} <= 10^n.

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