This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342841 #76 Jan 06 2025 21:06:15 %S A342841 1,841,832693,832046137,831916552903,831908477106883, %T A342841 831907430687799769,831907383078281024371,831907373418800027750413, %U A342841 831907372722449100147414487,831907372589073124899487831735,831907372581823023465031521920149,831907372580768386561159867257319711 %N A342841 Number of ordered triples (x, y, z) with gcd(x, y, z) = 1 and 1 <= {x, y, z} <= 10^n. %D A342841 Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54. %H A342841 Chai Wah Wu, <a href="/A342841/b342841.txt">Table of n, a(n) for n = 0..15</a> %H A342841 Karl-Heinz Hofmann, <a href="/A342841/a342841.gif">An animation of the cube with n = 1</a>. %F A342841 Lim_{n->infinity} a(n)/10^(3*n) = 1/zeta(3) = 1/Apéry's constant. %F A342841 a(n) = A071778(10^n). %e A342841 For visualization, the set(x, y, z) is inscribed in a cube matrix. %e A342841 "o" stands for a gcd = 1. %e A342841 "." stands for a gcd > 1. %e A342841 . %e A342841 For n=1, the size of the cube matrix is 10 X 10 X 10: %e A342841 . %e A342841 / : : : : : : : : : : %e A342841 / 100 Sum (z = 1) %e A342841 z = 7 |/1 2 3 4 5 6 7 8 9 10 | %e A342841 --+--------------------- 75 Sum (z = 2) %e A342841 1 /| o o o o o o o o o o 10 | %e A342841 2/ | o o o o o o o o o o 10 91 Sum (z = 3) %e A342841 / 10 | %e A342841 z = 8 |/1 2 3 4 5 6 7 8 9 10 10 75 Sum (z = 4) %e A342841 --+--------------------- 10 / %e A342841 1 /| o o o o o o o o o o 10 10 96 Sum (z = 5) %e A342841 2/ | o . o . o . o . o . 5 9 / %e A342841 / 10 10 67 Sum (z = 6) %e A342841 z = 9 |/1 2 3 4 5 6 7 8 9 10 5 10 / %e A342841 --+--------------------- 10 10 / %e A342841 1 /| o o o o o o o o o o 10 5 --/ %e A342841 2/ | o o o o o o o o o o 10 10 99 Sum (z = 7) %e A342841 / 7 5 / %e A342841 z = 10 |/1 2 3 4 5 6 7 8 9 10 10 10 / %e A342841 --+--------------------- 10 5 / %e A342841 1 | o o o o o o o o o o 10 7 --/ %e A342841 2 | o . o . o . o . o . 5 10 75 Sum (z = 8) %e A342841 3 | o o o o o o o o o o 10 10 / %e A342841 4 | o . o . o . o . o . 5 7 / %e A342841 5 | o o o o . o o o o . 8 10 / %e A342841 6 | o . o . o . o . o . 5 --/ %e A342841 7 | o o o o o o o o o o 10 91 Sum (z = 9) %e A342841 8 | o . o . o . o . o . 5 / %e A342841 9 | o o o o o o o o o o 10 / %e A342841 10 | o . o . . . o . o . 4 / %e A342841 --/ %e A342841 72 Sum (z = 10) %e A342841 / %e A342841 | %e A342841 ------ %e A342841 841 Cube Sum (z = 1..10) %o A342841 (Python) %o A342841 import math %o A342841 for n in range (0, 10): %o A342841 counter = 0 %o A342841 for x in range (1, pow(10, n)+1): %o A342841 for y in range(1, pow(10, n)+1): %o A342841 for z in range(1, pow(10, n)+1): %o A342841 if math.gcd(math.gcd(y, x),z) == 1: %o A342841 counter += 1 %o A342841 print(n, counter) %Y A342841 Cf. A342586 (for 10^n X 10^n), A018805, A002117 (zeta(3)), A071778. %Y A342841 Related counts of k-tuples: %Y A342841 pairs: A018805, A342632, A342586; %Y A342841 triples: A071778, A342935, A342841; %Y A342841 quadruples: A082540, A343527, A343193; %Y A342841 5-tuples: A343282; %Y A342841 6-tuples: A343978, A344038. - _N. J. A. Sloane_, Jun 13 2021 %K A342841 nonn,hard %O A342841 0,2 %A A342841 _Karl-Heinz Hofmann_, Mar 24 2021 %E A342841 a(5)-a(10) from _Hugo Pfoertner_, Mar 25 2021 %E A342841 a(11) from _Hugo Pfoertner_, Mar 26 2021 %E A342841 a(12) from _Bruce Garner_, Mar 29 2021