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A082540
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Number of ordered quadruples (a,b,c,d) with gcd(a,b,c,d)=1 (1 <= {a,b,c,d} <= n).
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5
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1, 15, 79, 239, 607, 1199, 2303, 3823, 6223, 9279, 13919, 19183, 27007, 35743, 47519, 60735, 78719, 97103, 122447, 148527, 181839, 216959, 262543, 306863, 365343, 423855, 495855, 569055, 661679, 748527, 862047, 972191, 1104831, 1237247
(list;
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OFFSET
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1,2
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LINKS
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Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^4.
a(n) is asymptotic to c*n^4 with c=0.92393....
Lim_{n->infinity} a(n)/n^4 = 1/zeta(4) = A215267 = 90/Pi^4. - Karl-Heinz Hofmann, Apr 11 2021
Lim_{n->infinity} n^4/a(n) = zeta(4) = A013662 = Pi^4/90. - Karl-Heinz Hofmann, Apr 11 2021
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PROG
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(PARI) a(n)=sum(k=1, n, moebius(k)*floor(n/k)^4)
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A082540(n):
if n == 0:
return 0
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (j2-j)*A082540(k1)
j, k1 = j2, n//j2
return n*(n**3-1)-c+j # Chai Wah Wu, Mar 29 2021
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CROSSREFS
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Sequence in context: A044583 A212746 A212741 * A269657 A189922 A085808
Adjacent sequences: A082537 A082538 A082539 * A082541 A082542 A082543
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KEYWORD
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nonn,changed
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AUTHOR
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Benoit Cloitre, May 11 2003
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STATUS
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approved
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