A394287 Decimal expansion of Product_{p prime} (1 - 2*p/(p^2+p+1)^2).

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

Offset

0,1

Comments

The asymptotic probability that the greatest common divisor of two cubefree numbers (A004709) selected independently at random equals their greatest common unitary divisor.

In general, the asymptotic probability that the greatest common divisor of two k-free numbers (numbers that are not divisible by a k-th power other than 1) selected independently at random equals their greatest common unitary divisor is P(k) = Product_{p prime} (1 - 2*p*(p^(k-1)-1)*(p^(k-2)-1)/((p+1)*(p^k-1)^2)). P(2) = 1, P(3) is this constant, and lim_{k->oo} P(k) = A394286.

Links

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

Formula

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

Example

0.870048743499707565237675464209844598664454651866249...

Prog

(PARI) prodeulerrat(1 - 2*p/(p^2+p+1)^2)

Crossrefs

Cf. A002117, A004709, A077610, A183030, A394286.

Sequence in context: A264587 A265024 A051762 * A247017 A330142 A198112

Adjacent sequences: A394284 A394285 A394286 * A394288 A394289 A394290

Keyword

nonn,cons

Author

Amiram Eldar, Mar 15 2026

Status

approved