This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A117507 #9 Feb 16 2025 08:33:00 %S A117507 2,23,3919,1400972,1332221503,2440266733544,9013120937567806, %T A117507 47710925260763230958,503649376979113850651329, %U A117507 5954610779280903922363948937,114594038963707117577230115067496 %N A117507 Numerators of partial sums of the Brun series divided by 4. %C A117507 The Brun series is the sum over reciprocals of the (odd) twin primes (see the mathworld link). %C A117507 The denominators divided by 5 are given in A117508. %C A117507 A001359 gives the lesser of the twin primes (offset 1). %C A117507 A006512 gives the greater of the twin primes (offset 1). %C A117507 A029707=[2,3,5,7,10,..] gives the indices for the lesser of the (odd) twin primes (offset 0). %C A117507 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. %H A117507 W. Lang: <a href="/A117507/a117507.txt">Rationals and more.</a> %H A117507 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BrunsConstant.html">Brun's Constant</a> %F A117507 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). %e A117507 Rationals 4*A117507(n)/5*A117508(n): 8/15, 92/105, 15676/15015, %e A117507 5603888/4849845, 5328886012/4360010655,... %K A117507 nonn,easy,frac %O A117507 1,1 %A A117507 _Wolfdieter Lang_, Apr 13 2006