A117508 -id:A117508 - cp's OEIS frontend

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

Wolfdieter Lang, Apr 13 2006

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.

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

W. Lang: Rationals and more.
Eric Weisstein's World of Mathematics, Brun's Constant

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).

cp's OEIS Frontend

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

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