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A345288
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Decimal expansion of Product_{p primes} sqrt(1 + 1/(4*p*(p-1))).
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2
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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
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OFFSET
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1,3
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
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FORMULA
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Equals sqrt(Pi) * lim_{n->infinity} sqrt(log(n))/n * Sum_{k=1..n} 1/A034444(k).
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EXAMPLE
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1.09698311911435715092514804937509666493621167605436728776543452986946366666...
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MATHEMATICA
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$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}]
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PROG
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(PARI) sqrt(prodeulerrat(1 + 1/(4*p*(p-1)))) \\ Amiram Eldar, Jun 13 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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