A393892 - OEIS

%I #5 Mar 02 2026 10:03:53

%S 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,

%T 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,

%U 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

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

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

%C 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)).

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

%e 0.994059906006358858812968535964200127457153710694827...

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

%Y Cf. A004709, A013661, A077610, A343359.

%Y 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).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 02 2026