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A077761 Decimal expansion of Mertens' constant, which is the limit of Sum{1/p(i), i=1..k } - log(log(p(k))) as k goes to infinity, where p(i) is the i-th prime number. 4
2, 6, 1, 4, 9, 7, 2, 1, 2, 8, 4, 7, 6, 4, 2, 7, 8, 3, 7, 5, 5, 4, 2, 6, 8, 3, 8, 6, 0, 8, 6, 9, 5, 8, 5, 9, 0, 5, 1, 5, 6, 6, 6, 4, 8, 2, 6, 1, 1, 9, 9, 2, 0, 6, 1, 9, 2, 0, 6, 4, 2, 1, 3, 9, 2, 4, 9, 2, 4, 5, 1, 0, 8, 9, 7, 3, 6, 8, 2, 0, 9, 7, 1, 4, 1, 4, 2, 6, 3, 1, 4, 3, 4, 2, 4, 6, 6, 5, 1, 0, 5, 1, 6, 1, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Graham, Knuth & Patashnik incorrectly give this constant as 0.261972128. - Robert G. Wilson v, Dec 02 2005

REFERENCES

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2004, pp. 94-98

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation For Computer Science, Addison-Wesley, Reading, MA, 1989, p. 23.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, 1996, Section VII.28, p. 257.

LINKS

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

Chris Caldwell, The Prime Pages, There are infinitely many primes, but, how big of an infinity?

H. Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. - From N. J. A. Sloane, Jan 26 2013

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants

Pieter Moree, Mathematical constants

P. Sebah and X. Gourdon, Constants from number theory

Torsten Sillke, The Harmonic Numbers and Series.

M. B. Villarino, Mertens' proof of Mertens' Theorem

Eric Weisstein's World of Mathematics, Mertens Constant

Eric Weisstein's World of Mathematics, Prime Zeta Function

Eric Weisstein's World of Mathematics, Harmonic Series of Primes

FORMULA

a(n)=A001620-sum(n=2,3,..infinity) zeta_prime(n)/n where the zeta prime sequence is A085548, A085541, A085964, A085965, A085966 etc. (Sebah and Gourdon) - R. J. Mathar, Apr 29 2006

EXAMPLE

0.26149721284764278375542683860869585905156664826119920619206421392...

MATHEMATICA

$MaxExtraPrecision = 400; RealDigits[ N[EulerGamma + NSum[(MoebiusMu[m]/m)*Log[N[Zeta[m], 120]], {m, 2, 1000}, Method -> "EulerMaclaurin", AccuracyGoal -> 120, NSumTerms -> 1000, PrecisionGoal -> 120, WorkingPrecision -> 120] , 120]][[1, 1 ;; 105]] (* Jean-Fran├žois Alcover, Mar 16 2011 *)

CROSSREFS

Sequence in context: A192043 A154584 A129677 * A220406 A220794 A220959

Adjacent sequences:  A077758 A077759 A077760 * A077762 A077763 A077764

KEYWORD

cons,nonn

AUTHOR

T. D. Noe, Nov 14 2002

STATUS

approved

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Last modified September 17 07:07 EDT 2014. Contains 246836 sequences.