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

0,1

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].

Links

Table of n, a(n) for n=0..105.

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

Example

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

Maple

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

Prog

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

Crossrefs

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

Sequence in context: A274439 A280342 A275476 * A373058 A136695 A337115

Adjacent sequences: A185377 A185378 A185379 * A185381 A185382 A185383

Keyword

cons,nonn

Author

R. J. Mathar, Jan 21 2013

Extensions

More digits from Vaclav Kotesovec, Jan 13 2021

Status

approved