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%I #32 Apr 12 2021 05:03:13
%S 1,31,241,991,3091,7501,16531,31711,57781,96601,157651,240031,362491,
%T 519961,739201,1012441,1383721,1822711,2409241,3091441,3966301,
%U 4974751,6257461,7680781,9481681,11474941,13916191,16610371,19911151,23435191
%N Number of ordered quintuples (a,b,c,d,e) with gcd(a,b,c,d,e)=1 (1<= {a,b,c,d,e} <= n).
%H Karl-Heinz Hofmann, <a href="/A082544/b082544.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^5; a(n) is asymptotic to c*n^5 with c=0.9643....
%F Lim_{n->infinity} a(n)/n^5 = 1/zeta(5) = A343308. - _Karl-Heinz Hofmann_, Apr 11 2021
%F Lim_{n->infinity} n^5/a(n) = zeta(5) = A013663. - _Karl-Heinz Hofmann_, Apr 11 2021
%o (PARI) a(n)=sum(k=1,n,moebius(k)*floor(n/k)^5)
%o (Python)
%o from functools import lru_cache
%o @lru_cache(maxsize=None)
%o def A082544(n):
%o if n == 0:
%o return 0
%o c, j = 1, 2
%o k1 = n//j
%o while k1 > 1:
%o j2 = n//k1 + 1
%o c += (j2-j)*A082544(k1)
%o j, k1 = j2, n//j2
%o return n*(n**4-1)-c+j # _Chai Wah Wu_, Mar 29 2021
%Y Cf. A018805 (pairs), A071778 (triples), A082540 (quadruples).
%K nonn
%O 1,2
%A _Benoit Cloitre_, May 11 2003
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