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%I #21 Jun 13 2021 13:08:37
%S 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,
%T 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,
%U 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
%N Decimal expansion of Silverman's constant.
%C Named after Robert D. Silverman. - _Amiram Eldar_, Aug 20 2020
%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 161.
%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 182.
%D Robert D. Silverman, A Peculiar Sum, USENET sci.math.research newsgroup posting, Mar 27 1996.
%H Steven R. Finch, <a href="https://web.archive.org/web/20120229012039/http://algo.inria.fr/csolve/arth.pdf">Series Involving Arithmetic Functions</a>, 2007.
%H Dave Rusin, <a href="https://web.archive.org/web/20140427083120/https://www.math.niu.edu/~rusin/known-math/96/numtheor.series">Re: A peculiar sum</a>, sci.math.research, Mar 27 1996.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SilvermanConstant.html">Silverman Constant</a>.
%H Paul Zimmermann, <a href="https://web.archive.org/web/20011123043051/http://www.mathsoft.com:80/asolve/constant/totient/zimmermn.html">Re: A Peculiar Sum</a> USENET sci.math.research newsgroup posting, Mar 29 1996.
%F From _Amiram Eldar_, Aug 20 2020: (Start)
%F Equals Sum_{k>=1} 1/(phi(k)*sigma(k)) = Sum_{k>=1} 1/A062354(k).
%F Equals Product_{p prime} (1 + Sum_{k>=1} 1/(p^(2*k) - p^(k-1))). (End)
%e 1.786576459365922463458590475541315750312621902384243294901...
%p 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
%t Sum[1/(EulerPhi[n]DivisorSigma[1, n]), {n, Infinity}]
%t $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 *)
%Y Cf. A000010, A000203, A062354.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Apr 16 2004
%E 37 more digits from _R. J. Mathar_, Jul 28 2010
%E More terms from _Vaclav Kotesovec_, Jun 13 2021
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