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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).

%I A343978 #44 Sep 13 2024 06:51:13
%S A343978 1,63,727,4031,15559,45863,116855,257983,526615,983583,1755143,
%T A343978 2935231,4776055,7407727,11256623,16498719,23859071,33434063,46467719,
%U A343978 62949975,84644439,111486599,146142583,187854119,240880239,303814503,382049919,473813703,586746719
%N 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).
%D A343978 Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.
%H A343978 Karl-Heinz Hofmann, <a href="/A343978/b343978.txt">Table of n, a(n) for n = 1..10000</a>
%H A343978 Joachim von zur Gathen and Jürgen Gerhard, <a href="/A342586/a342586_1.pdf">Extract from "3.4. (Non-)Uniqueness of the gcd" chapter</a>, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.
%F A343978 a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^6.
%F A343978 Lim_{n->infinity} a(n)/n^6 = 1/zeta(6) = A343359 = 945/Pi^6.
%F A343978 a(n) = n^6 - Sum_{k=2..n} a(floor(n/k)). - _Seiichi Manyama_, Sep 13 2024
%o A343978 (Python)
%o A343978 from labmath import mobius
%o A343978 def A343978(n): return sum(mobius(k)*(n//k)**6 for k in range(1, n+1))
%o A343978 (PARI) a(n)={sum(k=1, n+1, moebius(k)*(n\k)^6)} \\ _Andrew Howroyd_, May 08 2021
%o A343978 (Python)
%o A343978 from functools import lru_cache
%o A343978 @lru_cache(maxsize=None)
%o A343978 def A343978(n):
%o A343978     if n == 0:
%o A343978         return 0
%o A343978     c, j, k1 = 1, 2, n//2
%o A343978     while k1 > 1:
%o A343978         j2 = n//k1 + 1
%o A343978         c += (j2-j)*A343978(k1)
%o A343978         j, k1 = j2, n//j2
%o A343978     return n*(n**5-1)-c+j # _Chai Wah Wu_, May 17 2021
%Y A343978 Column k=6 of A344527.
%Y A343978 Cf. A343359, A013664, A018805, A071778, A082540, A082544.
%Y A343978 Cf. A342632, A342586, A342935, A342841, A343527, A343193.
%K A343978 nonn,less
%O A343978 1,2
%A A343978 _Karl-Heinz Hofmann_, May 06 2021
%E A343978 Edited by _N. J. A. Sloane_, Jun 13 2021