A065486 Decimal expansion of Product_{p prime} (1 + 1/(p+1)^2).

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

Offset

1,2

Comments

The constant c in the conjectured formula Sum_{a,b <= x} f(a,b) ~ c * x^2, where f(a, b) = gcd(ab, a + b)/gcd(a, b) (Tóth, 2026, p. 5, eq. (11)). - Amiram Eldar, Jun 24 2026

Links

Table of n, a(n) for n=1..99.

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

László Tóth, On the asymptotic density of the ordered pairs (a,b) of positive integers such that gcd(a*b,a+b) = gcd(a,b), arXiv:2606.20057 [math.NT], 2026. See p. 5 (Theorem 9).

Formula

Equals Sum_{k>=1} mu(k)^2/sigma(k)^2, where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022

Equals lim_{m->oo} (1/m^2) * Sum_{i,j=1..m} A000005(gcud(i, j)), where gcud is the greatest common unitary divisor function. - Amiram Eldar, Mar 14 2026

Example

1.26655850147152857161454711262964...

Mathematica

$MaxExtraPrecision = 800; digits = 99; terms = 800; P[n_] := PrimeZetaP[n]; LR = LinearRecurrence[{-3, -4, -2}, {0, 0, 2}, terms + 10]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

Prog

(PARI) prodeulerrat(1 + 1/(p+1)^2) \\ Amiram Eldar, Mar 15 2021

Crossrefs

Cf. A000005, A000203, A008683, A067009.

Sequence in context: A110388 A007507 A350031 * A069806 A123945 A291793

Adjacent sequences: A065483 A065484 A065485 * A065487 A065488 A065489

Keyword

cons,nonn

Author

N. J. A. Sloane, Nov 19 2001

Status

approved