A394289 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.

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, 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, 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

Offset

0,1

Comments

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).

Links

Table of n, a(n) for n=0..104.

Formula

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.

Example

0.958024304885093797499370832030193894313158894884929...

Prog

(PARI) c(m) = prodeulerrat((1-1/p^2) + (p-1)^2/p^2 * sum(k = 1, m, moebius(k)/(p^k-1)^2));
{my(c1 = 0, c2 = 1, m = 2); while(c2 != c1, c1 = c2; c2 = c(m); while(!issquarefree(m++), m++)); c2}

Crossrefs

Cf. A008683, A049599, A256936.

Sequence in context: A382012 A117019 A155692 * A011203 A245292 A203081

Adjacent sequences: A394286 A394287 A394288 * A394290 A394291 A394292

Keyword

nonn,cons

Author

Amiram Eldar, Mar 15 2026

Status

approved