A175549 Number of triples (a, b, c) with gcd(a, b, c) = 1 and -n <= a,b,c <= n.

0, 26, 98, 290, 578, 1154, 1730, 2882, 4034, 5762, 7490, 10370, 12674, 16706, 20162, 24770, 29378, 36290, 41474, 50114, 57026, 66242, 74882, 87554, 96770, 111170, 123266, 138818, 152642, 172802, 186626, 209666, 228098, 251138, 271874, 299522

Offset: 0

Author

Charles R Greathouse IV, Jun 24 2010

Keywords

• nonn

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A090025, A018805, A049691.

Programs

Mathematica

Table[If[n>0, 8 * Sum[MoebiusMu[k] * ((Floor[n/k] + 1)^3 - 1), {k, 1, n}] - 24 * Sum[EulerPhi[k], {k, 1, n}] - 6, 0], {n, 0, 35}] (* Indranil Ghosh, Mar 11 2017 *)

PARI

a(n)=if(n>0,8*sum(k=1,n,moebius(k)*((n\k+1)^3-1))-24*sum(k=1,n,eulerphi(k))-6)

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A175549(n):
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A175549(k1)
        j, k1 = j2, n//j2
    return 4*n*(n - 1)*(2*n + 5)-c+26*(j-1)# Chai Wah Wu, Mar 30 2021

Formula

For n > 0, a(n) = 8*A090025(n) - 12*A018805(n) - 18.

a(n) = 2*n*(4*n^2 + 6*n + 3) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

Extensions

Edited by Charles R Greathouse IV, Jul 19 2010