A345231 - OEIS

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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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..105.

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

Cf. A000005, A083281, A345288.

Sequence in context: A159894 A011502 A347185 * A335576 A270841 A097995

Adjacent sequences: A345228 A345229 A345230 * A345232 A345233 A345234

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Jun 11 2021

STATUS

approved