%I
%S 1,8,103,1916,58369,2474474,149329111,10799140436,1104684106441,
%T 119612262552092,16537863728067439,2484826470387072806,
%U 447865969660134667129,86094083836577627060684
%N Numerators of the partial sums of the reciprocals of the lower members of twin prime pairs.
%C The sum of the reciprocals of the lower bound twin primes is convergent. Proof: If S1 = 1/3 + 1/5 + 1/11 + 1/17 . . . is divergent then S1 + (S2 = 1/5 + 1/7 + 1/13 + 1/19 . . .) is divergent. But S1+S2 = 1/3+1/5 + 1/5+1/7 + 1/11+1/13 ... was proved to be convergent by V. Brun in 1919. So S1 is not divergent and since it is not oscillating, it is convergent as stated.
%e 1/3+1/5+1/11+1/17 = 1916/2805 and 1916 is the fourth entry in the table.
%t Accumulate[1/Select[Partition[Prime[Range[50]],2,1],#[[2]]#[[1]]==2&][[All,1]]]//Numerator (* _Harvey P. Dale_, Jul 24 2017 *)
%o (PARI) \Sum of the reciprocals of lower bound of twin primes { p=1; for(y=1,n, z=sum(x=1,y,1/twin[x]^p); print1(numerator(z)",") ); print(); print(z+.0); } \Build a twin prime table of lower bounds. Run only once in a session savetwins(n) = { twin = vector(n); c=1; forprime(x=3,n*10, if(isprime(x+2), twin[c]=x; c++; ) ) }
%K easy,nonn
%O 3,2
%A _Cino Hilliard_, Jan 23 2004
