%I A086239
%S A086239 3,3,4,9,8,1,3,2,5,2,9,9,9,9,3,1,8,1,0,6,3,3,1,7,1,2,1,4,8,7,5,4,3,5,7,
%T A086239 3,7,7,9,9,7,5,3,8,0,7,5,5,0,7,7,0,4,8,1,0,8,0,2,0,5,7,8,8,4,5,2,2,2,8,
%U A086239 4,3,2,7,1,8,8,4,1,1,0,6,2,4,8,9,9,6,3,1,0,2,9,8,0,3,3,4,5,3,9,2,4,8,6
%N A086239 Decimal expansion of sum(c[k]/prime[k], k=2..infinity), where c[k]=-1 
               if p==1 (mod 4) and c[k]=+1 if p==3 (mod 4).
%C A086239 This is sum_{p prime, p>=3} -(-4/p)/p where (-4/.) is the Legendre symbol 
               and is equal to - L(1,(-4/.)) plus an absolutely convergent sum (and 
               therefore converges).
%D A086239 Henri Cohen, "High Precision Computation of Hardy-Littlewood Constants", 
               preprint (1991), http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi 
               [From M. F. Hasler (MHasler(AT)univ-ag.fr), Oct 29 2009]
%D A086239 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and 
               its Applications, vol. 94, Cambridge University Press, pp. 94-98
%H A086239 D. Broadhurst, <a href="http://tech.groups.yahoo.com/group/primenumbers/
               message/21083">post in primenumbers group</a>, Oct 29 2009 [From 
               M. F. Hasler (MHasler(AT)univ-ag.fr), Oct 29 2009]
%H A086239 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeSums.html">PrimeSums</a>
%e A086239 0.33498132529999...
%o A086239 (PARI) /* the given number of primes and terms in the sum yield over 
               105 correct digits */ { P=vector(15, k, (2-prime(k)%4)/prime(k)); 
               -sum(s=1,60, moebius(s)/s*log( prod( k=2, #P, 1-P[k]^s, if(s%2, if(s==1, 
               Pi/4, sumalt(k=0,(-1)^k/(2*k+1)^s)) ,zeta(s)*(1-1/2^s) ))), sum(k=2,
               #P, P[k], .))} \\ [From M. F. Hasler (MHasler(AT)univ-ag.fr), Oct 
               29 2009]
%Y A086239 Sequence in context: A019466 A111573 A049854 this_sequence A016605 A060372 
               A128036
%Y A086239 Adjacent sequences: A086236 A086237 A086238 this_sequence A086240 A086241 
               A086242
%K A086239 nonn,cons
%O A086239 0,1
%A A086239 Eric Weisstein (eric(AT)weisstein.com), Jul 13, 2003
%E A086239 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jun 10 2008
%E A086239 Corrected a(9) and example, added a(10)-a(104) following Broadhurst and 
               Cohen. - M. F. Hasler, Oct 29 2009