A345288 Decimal expansion of Product_{p primes} sqrt(1 + 1/(4*p*(p-1))).

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

Offset: 1

Author

Vaclav Kotesovec, Jun 13 2021

Keywords

• nonn
• cons

Examples

			1.09698311911435715092514804937509666493621167605436728776543452986946366666...
		

Links

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.

Crossrefs

Cf. A034444, A083281, A345231.

Programs

Mathematica

$MaxExtraPrecision = 1000; Clear[f]; f[p_] := Sqrt[1 + 1/(4*p*(p-1))]; Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; Print[f[2] * Exp[N[Sum[Indexed[cc, n] * (PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]], {m, 100, 500, 100}]

PARI

sqrt(prodeulerrat(1 + 1/(4*p*(p-1)))) \\ Amiram Eldar, Jun 13 2021

Formula

Equals sqrt(Pi) * lim_{n->infinity} sqrt(log(n))/n * Sum_{k=1..n} 1/A034444(k).