A393953 Decimal expansion of Product_{p prime} ((1 - 1/(p+1)^2) * (1 - 1/(p^2+1)^2)).

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, 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, 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

Offset

0,1

Comments

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

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).

Links

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

Formula

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

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

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

Example

0.735919176675180664442827052201056739215767277163158...

Prog

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

Crossrefs

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

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).

Sequence in context: A175452 A084714 A340820 * A256779 A360094 A030760

Adjacent sequences: A393950 A393951 A393952 * A393954 A393955 A393956

Keyword

nonn,cons

Author

Amiram Eldar, Mar 04 2026

Status

approved