A394289 - OEIS

%I #9 Mar 17 2026 03:11:39

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

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

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

%N Decimal expansion of the asymptotic probability that the p-adic valuations of two positive integers selected independently at random are coprime for every prime p dividing both integers.

%C Equivalently, the asymptotic probability that the greatest common (1+e)-divisor of two positive integers selected independently at random is squarefree (see A049599 for the definition of a (1+e)-divisor).

%F Equals Product_{p prime} (1 - 1/p^2 + (1 - 1/p)^2 * Sum_{k>=1} mu(k)/(p^k-1)^2), where mu is the Moebius function.

%e 0.958024304885093797499370832030193894313158894884929...

%o (PARI) c(m) = prodeulerrat((1-1/p^2) + (p-1)^2/p^2 * sum(k = 1, m, moebius(k)/(p^k-1)^2));

%o {my(c1 = 0, c2 = 1, m = 2); while(c2 != c1, c1 = c2; c2 = c(m); while(!issquarefree(m++), m++)); c2}

%Y Cf. A008683, A049599, A256936.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 15 2026