%I #12 Mar 10 2026 13:34:36

%S 9,6,3,7,6,4,0,8,1,4,6,0,0,7,7,1,5,4,1,0,8,6,2,1,5,4,5,4,4,4,5,3,3,2,

%T 1,6,7,6,2,1,2,3,2,2,1,6,4,4,1,0,8,1,4,2,2,2,9,1,3,8,7,3,5,4,5,7,3,6,

%U 6,5,0,7,9,3,8,1,0,3,7,8,1,5,0,4,7,6,7,8,8,8,1,7,5,4,1,1,1,2,5,6,9,3,1,0,2

%N Decimal expansion of Product_{p prime} (1 - (p-1)^3/(p^3*(p^3-1))).

%C The asymptotic probability that the greatest common unitary divisor of three positive integers selected independently at random equals 1 (Tóth, 2014).

%C In general, the asymptotic probability that the greatest common unitary divisor of k positive integers selected independently at random equals 1 is Product_{p prime} (1 - (p-1)^k/(p^k*(p^k-1))).

%H László Tóth, <a href="https://doi.org/10.1007/978-1-4939-1106-6_19">Multiplicative arithmetic functions of several variables: a survey</a>, in: T. Rassias and P. Pardalos (eds.), Mathematics Without Boundaries: Surveys in Pure Mathematics, New York, NY: Springer New York, 2014, pp. 483-514; <a href="https://arxiv.org/abs/1310.7053">arXiv preprint</a>, arXiv:1310.7053 [math.NT], 2013-2014. See Proposition 22, p. 507.

%F Equals zeta(3) * Product_{p prime} (1 - 2/p^3 + 3/p^4 - 3/p^5 + 1/p^6).

%e 0.963764081460077154108621545444533216762123221644108...

%o (PARI) prodeulerrat(1 - (p-1)^3/(p^3*(p^3-1)))

%Y Cf. A002117, A077610, A306071, A319593.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 10 2026