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A093827 Decimal expansion of Silverman's constant. 3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Named after Robert D. Silverman. - Amiram Eldar, Aug 20 2020

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

Table of n, a(n) for n=1..88.

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.

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)

EXAMPLE

1.78657645...

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 *)

CROSSREFS

Cf. A000010, A000203, A062354.

Sequence in context: A303985 A242816 A333566 * A245736 A088660 A244263

Adjacent sequences:  A093824 A093825 A093826 * A093828 A093829 A093830

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Apr 16 2004

EXTENSIONS

37 more digits from R. J. Mathar, Jul 28 2010

STATUS

approved

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Last modified October 25 03:54 EDT 2020. Contains 338011 sequences. (Running on oeis4.)