A117507 Numerators of partial sums of the Brun series divided by 4.

2, 23, 3919, 1400972, 1332221503, 2440266733544, 9013120937567806, 47710925260763230958, 503649376979113850651329, 5954610779280903922363948937, 114594038963707117577230115067496

Offset

1,1

Comments

The Brun series is the sum over reciprocals of the (odd) twin primes (see the mathworld link).

The denominators divided by 5 are given in A117508.

A001359 gives the lesser of the twin primes (offset 1).

A006512 gives the greater of the twin primes (offset 1).

A029707=[2,3,5,7,10,..] gives the indices for the lesser of the (odd) twin primes (offset 0).

The proof that the partial sums of the Brun series have numerators divisible by 4 and denominators divisible by 5 can be given by induction.

Links

Table of n, a(n) for n=1..11.

W. Lang: Rationals and more.

Eric Weisstein's World of Mathematics, Brun's Constant

Formula

a(n)=numerator(r(n))/4, with r(n):=sum(1/ltp(k) + 1/(ltp(k)+2),k=1..n), n>=1, with ltp(k):=A001359(k) (lesser twin primes).

Example

Rationals 4*A117507(n)/5*A117508(n): 8/15, 92/105, 15676/15015,
5603888/4849845, 5328886012/4360010655,...

Crossrefs

Sequence in context: A081781 A088734 A087388 * A068656 A030997 A090511

Adjacent sequences: A117504 A117505 A117506 * A117508 A117509 A117510

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Apr 13 2006

Status

approved