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

%I #20 Mar 31 2021 10:47:00

%S 0,26,98,290,578,1154,1730,2882,4034,5762,7490,10370,12674,16706,

%T 20162,24770,29378,36290,41474,50114,57026,66242,74882,87554,96770,

%U 111170,123266,138818,152642,172802,186626,209666,228098,251138,271874,299522

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

%H Indranil Ghosh, <a href="/A175549/b175549.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%t 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 *)

%o (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)

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A175549(n):

%o if n == 0:

%o return 0

%o c, j = 0, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

%o c += (j2-j)*A175549(k1)

%o j, k1 = j2, n//j2

%o return 4*n*(n - 1)*(2*n + 5)-c+26*(j-1)# _Chai Wah Wu_, Mar 30 2021

%Y Cf. A090025, A018805, A049691.

%K nonn

%O 0,2

%A _Charles R Greathouse IV_, Jun 24 2010

%E Edited by _Charles R Greathouse IV_, Jul 19 2010