Rationals  4*A117507(n)/5*A117508(n), n=1,...,20.  
   
  Brun's (convergent) series is (the sum over the reciprocal of each member of the odd prime twins).

  B:= (1/3+1/5)+(1/5+1/7)+(1/11+1/13)+(1/17+1/19)+ ...
 
  The partial sums are B_n:=sum(1/ltp(k) + 1/(ltp(k)+2),k=1..n) with ltp(k):=prime(A029707(k-1)), where 

  A029707=[2,3,5,7,10,13,17,20,26,28,33,35,41,43,45,49,52,57,60,64,69,81,83,,...] (offset 0), therefore,
  
  ltp(k)= A001359(k) = [3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281...], k>=1.

  Hence B_1 = 1/prime(2) + 1/prime(3) = 1/3 + 1/5 = 8/15. 

  The sequence of the partial sums B_n is, for n=1..20:

  n=1:   8/15,  n=2:  92/105,  n=3:  15676/15015,  n=4: 5603888/4849845,  n=5: 5328886012 / 4360010655,  n=6: 9761066934176 / 7686698784765, 

  n=7:   36052483750271224 / 27664428926369235,  n=8: 190843701043052923832 / 143384735125371745005, 

  n=9:   2014597507916455402605316 / 1491631399509242263287015,  n=10: 23818443117123615689455795748 / 17396897012476292516716455945,

  n=11:  458376155854828470308920460269984 / 331289109808586038395831470560635,

  n=12:  10412391863520361564999150876782560516 / 7453673681583377277867812256143726865, 

  n=13:  340034406511566210164439901669089919829284 / 241491573609619840425639249286800606699135,
  
  n=14:  12627421091501959224015193566953793147639363932 / 8902103877971416177610339646459330764750213505, 

  n=15:  498558022185827988265401327905287447749747068774176 / 348989178328113428410858145160145143970502620036515,

  n=16:  26075680732603516037355708542693583582025651070824641848 / 18141504457030320349081638959859825019018637697358159245,

  n=17:  1510641056656604474003210643051340436750224422123160462240552/1044932515220489421786753322448966061270454512730132614352755,

  n=18:  110688485947428873838124899461922108171740655585230546148624487948 / 76174535427058458358832530453207176900554863523513937453701486745,
 
  n=19:  8845242905978247304743587927086040655913245097131550582100952793612984 / 6057627580765969784069439319230394328662824411980398848130703330422635

  n=20:   864802439807038492530914409721536221629648619923152297194686606666852425752 / 
          589667641594501796690671431651844275135025316735407965073587054293330558805
 
  .
  .
  .
  #####################################################################################################################################################

  The even numerators are, for n=1..20: 
  
[8, 92, 15676, 5603888, 5328886012, 9761066934176, 36052483750271224, 190843701043052923832, 2014597507916455402605316, 
23818443117123615689455795748, 458376155854828470308920460269984, 10412391863520361564999150876782560516, 340034406511566210164439901669089919829284, 
12627421091501959224015193566953793147639363932, 498558022185827988265401327905287447749747068774176, 
26075680732603516037355708542693583582025651070824641848, 1510641056656604474003210643051340436750224422123160462240552,
 110688485947428873838124899461922108171740655585230546148624487948, 8845242905978247304743587927086040655913245097131550582100952793612984, 
864802439807038492530914409721536221629648619923152297194686606666852425752]

  Divided by 4, this gives A117507:

[2, 23, 3919, 1400972, 1332221503, 2440266733544, 9013120937567806, 47710925260763230958, 503649376979113850651329, 5954610779280903922363948937, 114594038963707117577230115067496, 2603097965880090391249787719195640129, 85008601627891552541109975417272479957321, 3156855272875489806003798391738448286909840983, 124639505546456997066350331976321861937436767193544, 6518920183150879009338927135673395895506412767706160462, 377660264164151118500802660762835109187556105530790115560138, 27672121486857218459531224865480527042935163896307636537156121987, 2211310726494561826185896981771510163978311274282887645525238198403246, 216200609951759623132728602430384055407412154980788074298671651666713106438]
 
 ####################################################################################

 The denominators are, for n=1..20: 

 [15, 105, 15015, 4849845, 4360010655, 7686698784765, 27664428926369235, 143384735125371745005, 1491631399509242263287015, 
17396897012476292516716455945, 331289109808586038395831470560635, 7453673681583377277867812256143726865, 241491573609619840425639249286800606699135,
 8902103877971416177610339646459330764750213505, 348989178328113428410858145160145143970502620036515, 
18141504457030320349081638959859825019018637697358159245, 1044932515220489421786753322448966061270454512730132614352755, 
76174535427058458358832530453207176900554863523513937453701486745, 6057627580765969784069439319230394328662824411980398848130703330422635, 
589667641594501796690671431651844275135025316735407965073587054293330558805]


 divided by 5 this gives A117508(n), n=1..20:

[3, 21, 3003, 969969, 872002131, 1537339756953, 5532885785273847, 28676947025074349001, 298326279901848452657403,
 3479379402495258503343291189, 66257821961717207679166294112127, 1490734736316675455573562451228745373, 48298314721923968085127849857360121339827, 
1780420775594283235522067929291866152950042701, 69797835665622685682171629032029028794100524007303, 
3628300891406064069816327791971965003803727539471631849, 208986503044097884357350664489793212254090902546026522870551,
 15234907085411691671766506090641435380110972704702787490740297349, 1211525516153193956813887863846078865732564882396079769626140666084527, 
117933528318900359338134286330368855027005063347081593014717410858666111761]


#######################################################################################

 The values for r(n)= 4*A117507(n)/5*A117508(n), n=1..20,  are (Maple10 10digits):

  [.5333333333, .8761904762, 1.044022644, 1.155477752, 1.222218576, 1.269864633, 1.303207229, 1.330990366, 1.350600094, 1.369120200,
 1.383613715, 1.396947641, 1.408059095, 1.418476044, 1.428577312, 1.437349410, 1.445682888, 1.453090397, 1.460182685, 1.466593007]

 ########################################################################################################################################################

 According to the given mathworld link Brun's constant is approximately 1.902160583104 (Sebah 2002). 

############################################################# e.o.f.####################################################################################