A085609 Decimal expansion of Sum{p prime>=2} log(p)/(p^2-p+1).

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

Offset

0,1

Comments

Appears in the asymptotic formula for Sum{k=1..n} 1/phi(k), with phi(k) being Euler's totient function. - Stanislav Sykora, Nov 14 2014

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 116.

Links

Table of n, a(n) for n=0..104.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z2).

X. Gourdon, P. Sebah, Some Constants from Number theory.

Wikipedia, Euler's totient function

Formula

Equals lim_{n->infinity} (Gamma + log(n) - c*Sum_{k=1..n} 1/phi(k)), where Gamma is the Euler-Mascheroni constant, and c = zeta(6)/(zeta(2)*zeta(3)) = 1/A082695. This equals further lim_{n->infinity} Sum{k=1..n} (1/k - c/phi(k)) and lim_{n->infinity}(A001008(n)/A002805(n) - (A028415(n)/A048049(n))/A082695). - Stanislav Sykora, Nov 15 2014

Example

0.60838171786332472...

Mathematica

digits = 105; m0 = 100; dm = 100; Clear[s]; s[n_] := s[n] = Sum[ Switch[ Mod[k, 6], 0, 1, 1, 0, 2, -1, 3, -1, 4, 0, 5, 1] * PrimeZetaP'[k], {k, 2, n}] // N[#, digits+40]&; Print[m0, " ", s[m0]]; s[m = m0+dm]; While[ Print[m, " ", s[m]]; RealDigits[s[m], 10, digits+5] != RealDigits[s[m-dm], 10, digits+5], m = m+dm]; RealDigits[s[m], 10, digits] // First (* Jean-François Alcover, Sep 11 2015 *)

Crossrefs

Cf. A001008, A002805, A028415, A048049, A082695.

Sequence in context: A010491 A257095 A159845 * A153609 A156015 A201898

Adjacent sequences: A085606 A085607 A085608 * A085610 A085611 A085612

Keyword

cons,nonn

Author

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

Extensions

More terms from Benoit Cloitre, Mar 06 2013

More digits from Jean-François Alcover, Sep 11 2015

Status

approved