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
- Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54. (See link below.)
Related counts of k-tuples:
-
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
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
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.
Related counts of k-tuples:
-
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)
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
(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.
Related counts of k-tuples:
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
Related counts of k-tuples:
-
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))
Showing 1-4 of 4 results.
