A086241 - OEIS

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A086241

Decimal expansion of value to which Sum_{k>=2} d(k)/prime(k) appears to converge, where d(k)=-1 if p mod 3 = 1, d(k)=+1 if p mod 3 = 2 and d(k)=0 if p mod 3 = 0.

6, 4, 1, 9, 4, 4, 8, 3, 8, 5, 3, 3, 1, 9, 5, 7, 0, 8, 6, 6, 1, 3, 9, 2, 6, 3, 9, 7, 2, 1, 7, 3, 4, 3, 1, 6, 6, 7, 5, 4, 1, 1, 0, 4, 4, 0, 1, 5, 8, 8, 9, 6, 5, 4, 9, 0, 8, 1, 7, 0, 8, 4, 5, 1, 3, 1, 7, 3, 3, 2, 8, 2, 6, 9, 0, 7, 3, 7, 2, 3, 3, 5, 9, 8, 1, 7, 4, 1, 5, 9, 9, 4, 5, 6, 0, 6, 5, 7

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OFFSET

0,1

COMMENTS

It is not known if this series actually converges.

REFERENCES

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.

LINKS

Table of n, a(n) for n=0..97.

Richard J. Mathar, Table of Dirichlet L-series..., arXiv:1008.2547 [math.NT], 2010-2015, value of S(m=3,r=2,s=1).

Eric Weisstein's World of Mathematics, Prime Sums.

FORMULA

Equals A161529 + A368644. - Amiram Eldar, Jan 02 2024

EXAMPLE

0.64194483853319570866139263972173431667541104401588965490817...

MATHEMATICA

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[3, 2, 1], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *)

CROSSREFS

Cf. A161529, A368644.

Sequence in context: A324034 A365319 A343614 * A378412 A204023 A360984

Adjacent sequences: A086238 A086239 A086240 * A086242 A086243 A086244

KEYWORD

nonn,cons,hard

AUTHOR

Eric W. Weisstein, Jul 13 2003

EXTENSIONS

More digits from R. J. Mathar, Jul 28 2010

Sign typo in definition corrected by R. J. Mathar, Aug 01 2010

STATUS

approved