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A345231
Decimal expansion of 1/sqrt(Pi) * Product_{p primes} sqrt(p*(p-1)) * log(p/(p-1)).
2
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
0,1
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.
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).
EXAMPLE
0.5468559552804744668455171009907617899102104859297429478286893714993514862739...
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}]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Jun 11 2021
STATUS
approved