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

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, 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, 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

Offset: 0

Author

R. J. Mathar, Jan 21 2013

Keywords

• cons
• nonn

Comments

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].

Examples

			0.263672061761153178749842188233776 .. = 1/(2*4) +1/(3*5) + 1/(5*7) + 1/(7*9) + 1/(11*13)+ ...
		

Crossrefs

Cf. A136141 (1/(p(p-1))), A179119 (1/(p(p+1))).

Programs

Maple

read("transforms") ;
Digits := 300 ;
# insert coding of ZetaM(s,M) and Hurw(a) from A179119 here...
A185380 := proc()
        Hurw(2) ;
end proc:
A185380() ;

PARI

sumeulerrat(1/(p*(p+2))) \\ Amiram Eldar, Mar 19 2021

Formula

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

Extensions

More digits from Vaclav Kotesovec, Jan 13 2021