A345231 Decimal expansion of 1/sqrt(Pi) * Product_{p primes} sqrt(p*(p-1)) * log(p/(p-1)).

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

Offset: 0

Author

Vaclav Kotesovec, Jun 11 2021

Keywords

• nonn
• cons

Examples

			0.5468559552804744668455171009907617899102104859297429478286893714993514862739...
		

Links

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
• Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (7), constant A1.

Crossrefs

Cf. A000005, A083281, A345288.

Programs

Mathematica

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

Formula

Equals lim_{n->infinity} sqrt(log(n))/n * Sum_{k=1..n} 1/d(k), where d(n) = A000005(n).

Equals A083281/sqrt(Pi).