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A093827
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Decimal expansion of Silverman's constant.
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3
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1, 7, 8, 6, 5, 7, 6, 4, 5, 9, 3, 6, 5, 9, 2, 2, 4, 6, 3, 4, 5, 8, 5, 9, 0, 4, 7, 5, 5, 4, 1, 3, 1, 5, 7, 5, 0, 3, 1, 2, 6, 2, 1, 9, 0, 2, 3, 8, 4, 2, 4, 3, 2, 9, 4, 9, 0, 1, 0, 7, 2, 4, 9, 6, 2, 1, 4, 2, 4, 5, 2, 7, 9, 1, 3, 4, 7, 8, 6, 2, 2, 3, 7, 7, 3, 2, 6, 9, 2, 4, 3, 9, 0, 3, 2, 8, 0, 5, 6, 8, 7, 6, 9, 0, 2
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OFFSET
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1,2
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COMMENTS
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Named after Robert D. Silverman. - Amiram Eldar, Aug 20 2020
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REFERENCES
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Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 161.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 182.
Robert D. Silverman, A Peculiar Sum, USENET sci.math.research newsgroup posting, Mar 27 1996.
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LINKS
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Paul Zimmermann, Re: A Peculiar Sum USENET sci.math.research newsgroup posting, Mar 29 1996.
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FORMULA
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Equals Sum_{k>=1} 1/(phi(k)*sigma(k)) = Sum_{k>=1} 1/A062354(k).
Equals Product_{p prime} (1 + Sum_{k>=1} 1/(p^(2*k) - p^(k-1))). (End)
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EXAMPLE
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1.786576459365922463458590475541315750312621902384243294901...
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MAPLE
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read("transforms") ; Digits := 140 ; kmax := 450 ; tmax := kmax-10 ; 1+add(1/(p^(2*k)-p^(k-1)), k=1..kmax) : xt := subs(p=1/x, %) : xt := taylor(xt, x=0, tmax) ; L := [] ; for n from 1 to tmax-1 do L := [op(L), coeftayl(xt, x=0, n)]; end do: Le := EULERi(L) ; x := 1.0 ; for i from 2 to nops(Le) do x := x*Zeta(i)^op(i, Le) ; x := evalf(x) ; print(x) ; end do: # R. J. Mathar, Jul 28 2010
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MATHEMATICA
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Sum[1/(EulerPhi[n]DivisorSigma[1, n]), {n, Infinity}]
$MaxExtraPrecision = 500; m = 500; f[p_] := 1 + Sum[1/(p^(2*k) - p^(k - 1)), {k, 1, 2*m}]; c = Rest@CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Aug 20 2020 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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