A393892 Decimal expansion of Product_{p prime} (1 - (p-1)/(p^6*(p+1))).

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

Offset

0,1

Comments

The asymptotic probability that the greatest common unitary divisor of two positive integers selected at random is a cubefree number (A004709).

In general, the asymptotic probability that the greatest common unitary divisor of two positive integers selected at random is a k-free number (a number that is not divisible by a k-th power other than 1) is Product_{p prime} (1 - (p-1)/(p^(2*k)*(p+1))) = zeta(2) * Product_{p prime} (1 - 1/p^2 - 1/p^(2*k) + 2/p^(2*k+1) - 1/p^(2*k+2)).

Links

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

Formula

Equals zeta(2) * Product_{p prime} (1 - 1/p^2 - 1/p^6 + 2/p^7 - 1/p^8).

Example

0.994059906006358858812968535964200127457153710694827...

Prog

(PARI) prodeulerrat(1 - (p-1)/(p^6*(p+1)))

Crossrefs

Cf. A004709, A013661, A077610, A343359.

The asymptotic probability that the greatest common unitary divisor of two positive integers selected at random is: A021016 (even), A306071 (1), A393891 (squarefree), this constant (cubefree), A393893 (powerful), A393894 (cubefull), A393895 (square), A393896 (cube), A393897 (exponentially odd number), A393898 (prime), A393899 (prime power), A393900 (perfect power).

Sequence in context: A172502 A118739 A192031 * A387453 A283749 A249023

Adjacent sequences: A393889 A393890 A393891 * A393893 A393894 A393895

Keyword

nonn,cons

Author

Amiram Eldar, Mar 02 2026

Status

approved