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).

1, 31, 241, 991, 3091, 7501, 16531, 31711, 57781, 96601, 157651, 240031, 362491, 519961, 739201, 1012441, 1383721, 1822711, 2409241, 3091441, 3966301, 4974751, 6257461, 7680781, 9481681, 11474941, 13916191, 16610371, 19911151, 23435191

Offset: 1

Author

Benoit Cloitre, May 11 2003

Keywords

• nonn

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Crossrefs

Column k=5 of A344527.

Cf. A018805 (pairs), A071778 (triples), A082540 (quadruples), A343978.

Cf. A015650.

Programs

PARI

a(n)=sum(k=1,n,moebius(k)*floor(n/k)^5)

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A082544(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082544(k1)
        j, k1 = j2, n//j2
    return n*(n**4-1)-c+j # Chai Wah Wu, Mar 29 2021

Formula

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^5; a(n) is asymptotic to c*n^5 with c=0.9643....

Lim_{n->infinity} a(n)/n^5 = 1/zeta(5) = A343308. - Karl-Heinz Hofmann, Apr 11 2021

Lim_{n->infinity} n^5/a(n) = zeta(5) = A013663. - Karl-Heinz Hofmann, Apr 11 2021

a(n) = n^5 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024