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

Author

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

Keywords

• cons
• nonn

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

Examples

			0.60838171786332472...
		

References

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

Links

• 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

Crossrefs

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

Programs

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 *)

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

Extensions

More terms from Benoit Cloitre, Mar 06 2013

More digits from Jean-François Alcover, Sep 11 2015