A093827 Decimal expansion of Silverman's constant.
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
Offset: 1
Examples
1.786576459365922463458590475541315750312621902384243294901...
References
- 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.
Links
- Steven R. Finch, Series Involving Arithmetic Functions, 2007.
- Dave Rusin, Re: A peculiar sum, sci.math.research, Mar 27 1996.
- Eric Weisstein's World of Mathematics, Silverman Constant.
- Paul Zimmermann, Re: A Peculiar Sum USENET sci.math.research newsgroup posting, Mar 29 1996.
Programs
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Maple
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 -
Mathematica
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 *)
Formula
From Amiram Eldar, Aug 20 2020: (Start)
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)
Extensions
37 more digits from R. J. Mathar, Jul 28 2010
More terms from Vaclav Kotesovec, Jun 13 2021
Comments