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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.
8
1, 841, 832693, 832046137, 831916552903, 831908477106883, 831907430687799769, 831907383078281024371, 831907373418800027750413, 831907372722449100147414487, 831907372589073124899487831735, 831907372581823023465031521920149, 831907372580768386561159867257319711
OFFSET
0,2
REFERENCES
Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.
FORMULA
Lim_{n->infinity} a(n)/10^(3*n) = 1/zeta(3) = 1/Apéry's constant.
a(n) = A071778(10^n).
EXAMPLE
For visualisation, 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)
PROG
(Python)
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)
CROSSREFS
Cf. A342586 (for 10^n X 10^n), A018805, A002117 (zeta(3)), A071778.
Related counts of k-tuples:
triples: A071778, A342935, A342841;
quadruples: A082540, A343527, A343193;
5-tuples: A343282;
6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021
Sequence in context: A253599 A121499 A253514 * A337730 A049530 A158404
KEYWORD
nonn,hard
AUTHOR
Karl-Heinz Hofmann, Mar 24 2021
EXTENSIONS
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
STATUS
approved