cp's OEIS Frontend

A082544 Number of ordered quintuples (a,b,c,d,e) with gcd(a,b,c,d,e)=1 (1<= {a,b,c,d,e} <= n).

%I A082544 #39 Sep 13 2024 06:51:05
%S A082544 1,31,241,991,3091,7501,16531,31711,57781,96601,157651,240031,362491,
%T A082544 519961,739201,1012441,1383721,1822711,2409241,3091441,3966301,
%U A082544 4974751,6257461,7680781,9481681,11474941,13916191,16610371,19911151,23435191
%N A082544 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 A082544 Karl-Heinz Hofmann, <a href="/A082544/b082544.txt">Table of n, a(n) for n = 1..10000</a>
%F A082544 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 A082544 Lim_{n->infinity} a(n)/n^5 = 1/zeta(5) = A343308. - _Karl-Heinz Hofmann_, Apr 11 2021
%F A082544 Lim_{n->infinity} n^5/a(n) =   zeta(5) = A013663. - _Karl-Heinz Hofmann_, Apr 11 2021
%F A082544 a(n) = n^5 - Sum_{k=2..n} a(floor(n/k)). - _Seiichi Manyama_, Sep 13 2024
%o A082544 (PARI) a(n)=sum(k=1,n,moebius(k)*floor(n/k)^5)
%o A082544 (Python)
%o A082544 from functools import lru_cache
%o A082544 @lru_cache(maxsize=None)
%o A082544 def A082544(n):
%o A082544     if n == 0:
%o A082544         return 0
%o A082544     c, j = 1, 2
%o A082544     k1 = n//j
%o A082544     while k1 > 1:
%o A082544         j2 = n//k1 + 1
%o A082544         c += (j2-j)*A082544(k1)
%o A082544         j, k1 = j2, n//j2
%o A082544     return n*(n**4-1)-c+j # _Chai Wah Wu_, Mar 29 2021
%Y A082544 Column k=5 of A344527.
%Y A082544 Cf. A018805 (pairs), A071778 (triples), A082540 (quadruples), A343978.
%Y A082544 Cf. A015650.
%K A082544 nonn
%O A082544 1,2
%A A082544 _Benoit Cloitre_, May 11 2003