A393953 - OEIS

%I #7 Mar 06 2026 08:32:41

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

%T 3,9,2,1,5,7,6,7,2,7,7,1,6,3,1,5,8,9,3,6,4,2,3,8,4,5,7,4,6,7,8,5,6,9,

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

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

%C The asymptotic probability that the greatest common infinitary divisor of two positive integers selected independently at random is a fourth power (A000583).

%C In general, the asymptotic probability that the greatest common infinitary divisor of two positive integers selected independently at random is a 2^m-power is Product_{p prime} Product_{k=0..m-1} (1 - 1/(p^(2^k)+1)^2).

%F Equals A065472 * Product_{p prime} (1 - 1/(p^2+1)^2).

%F Equals A065472 * zeta(4)^2 * Product_{p prime} (1 - 3/p^4 + 2/p^6).

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

%e 0.735919176675180664442827052201056739215767277163158...

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

%Y Cf. A000583, A013662, A065472, A077609, A183031.

%Y The asymptotic probability that the greatest common infinitary divisor of two positive integers selected independently at random is: A065472 (square), A393948 (1), A393949 (prime), A393950 (squarefree), A393951 (Fermi-Dirac prime), A393952 (exponentially 2^n-number), this constant (4th power).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 04 2026