A082540 -id:A082540 - cp's OEIS frontend

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

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

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).

Original entry on oeis.org

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

A082544 Number of ordered quintuples (a,b,c,d,e) with gcd(a,b,c,d,e)=1 (1<= {a,b,c,d,e} <= n).

Original entry on oeis.org

1, 31, 241, 991, 3091, 7501, 16531, 31711, 57781, 96601, 157651, 240031, 362491, 519961, 739201, 1012441, 1383721, 1822711, 2409241, 3091441, 3966301, 4974751, 6257461, 7680781, 9481681, 11474941, 13916191, 16610371, 19911151, 23435191

Offset: 1

Benoit Cloitre, May 11 2003

nonn

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

Column k=5 of A344527.
Cf. A018805 (pairs), A071778 (triples), A082540 (quadruples), A343978.
Cf. A015650.

a(n)=sum(k=1,n,moebius(k)*floor(n/k)^5)

from functools import lru_cache
@lru_cache(maxsize=None)
def A082544(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082544(k1)
        j, k1 = j2, n//j2
    return n*(n**4-1)-c+j # Chai Wah Wu, Mar 29 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^5; a(n) is asymptotic to c*n^5 with c=0.9643....
Lim_{n->infinity} a(n)/n^5 = 1/zeta(5) = A343308. - Karl-Heinz Hofmann, Apr 11 2021
Lim_{n->infinity} n^5/a(n) =   zeta(5) = A013663. - Karl-Heinz Hofmann, Apr 11 2021
a(n) = n^5 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024

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

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).

A344523 a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i,j,k,l).

Original entry on oeis.org

1, 17, 84, 276, 649, 1417, 2528, 4432, 7033, 10905, 15556, 22836, 30673, 41729, 54944, 71968, 89969, 115457, 140820, 175444, 212537, 257113, 302720, 366160, 426505, 500873, 580676, 677108, 769761, 895377, 1008928, 1153120, 1300417, 1469073, 1640020, 1860340, 2054921

Offset: 1

Seiichi Manyama, May 22 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=4 of A344479.

Cf. A082540, A343498, A344138, A344522, A344524, A344525.

a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)

a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, gcd([i, j, k, l])))));

a(n) = sum(k=1, n, eulerphi(k)*(n\k)^4);

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4)/(1-x))

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^4.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.
a(n) ~ 90 * zeta(3) * n^4 / Pi^4. - Vaclav Kotesovec, May 23 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

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Java

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

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).

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

Python

Python

Formula

Extensions

A082544 Number of ordered quintuples (a,b,c,d,e) with gcd(a,b,c,d,e)=1 (1<= {a,b,c,d,e} <= n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

PARI

Python

Python

Formula

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Python

Formula

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Python

Formula

Extensions

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

Views

Author