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A185380 Decimal expansion of sum 1/(p*(p+2)) over the primes p.

%I A185380 #14 Mar 19 2021 06:59:13
%S A185380 2,6,3,6,7,2,0,6,1,7,6,1,1,5,3,1,7,8,7,4,9,8,4,2,1,8,8,2,3,3,7,7,6,7,
%T A185380 5,3,0,8,7,4,9,6,3,1,8,3,9,6,7,5,6,8,0,2,1,2,2,2,3,8,1,2,6,8,3,2,2,4,
%U A185380 3,8,9,8,1,6,3,2,2,9,8,2,4,9,8,3,9,2,2,6,6,1,7,5,4,5,1,8,0,9,6,4,0,0,6,9,9,4
%N A185380 Decimal expansion of sum 1/(p*(p+2)) over the primes p.
%C A185380 If we omit the first term 1/(2*4)=0.125 from the sum, 0.138672... remains, which is an upper limit of A209329 in the sense that we "fake" prime gaps of 2 here [which are actually larger on average].
%F A185380 Equals -1/8 + Sum_{k>=2} (-1)^k * 2^(k-2) * P(k), where P is the prime zeta function. - _Vaclav Kotesovec_, Jan 13 2021
%e A185380 0.263672061761153178749842188233776 .. = 1/(2*4) +1/(3*5) + 1/(5*7) + 1/(7*9) + 1/(11*13)+ ...
%p A185380 read("transforms") ;
%p A185380 Digits := 300 ;
%p A185380 # insert coding of ZetaM(s,M) and Hurw(a) from A179119 here...
%p A185380 A185380 := proc()
%p A185380         Hurw(2) ;
%p A185380 end proc:
%p A185380 A185380() ;
%o A185380 (PARI) sumeulerrat(1/(p*(p+2))) \\ _Amiram Eldar_, Mar 19 2021
%Y A185380 Cf. A136141 (1/(p(p-1))), A179119 (1/(p(p+1))).
%K A185380 cons,nonn
%O A185380 0,1
%A A185380 _R. J. Mathar_, Jan 21 2013
%E A185380 More digits from _Vaclav Kotesovec_, Jan 13 2021