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 A071990 Numerators of the partial sums of the reciprocals of the upper members of twin prime pairs. 0
 1, 12, 191, 4084, 135249, 6083702, 382629607, 28634912196, 3000711370793, 332363027120752, 46774578929554863, 7143041842570860878, 1304982717560745380233, 254050342563254798982984 (list; graph; refs; listen; history; text; internal format)
 OFFSET 5,2 COMMENTS The sum of the reciprocals of the upper bound twin primes is convergent. Proof: If S2 = 1/5 + 1/7 + 1/13 + 1/19 . . . is divergent then S2 + (S1 = 1/3 + 1/5 + 1/11 + 1/17 . . .) 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 S2 is not divergent and since it is not oscillating, it is convergent as stated. LINKS EXAMPLE 1/5+1/7+1/13+1/19 = 4084/8645 and 4084 is the fourth entry in the table. PROG (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]+2)^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++; ) ) } CROSSREFS Sequence in context: A166773 A202632 A218886 * A196483 A196716 A086948 Adjacent sequences:  A071987 A071988 A071989 * A071991 A071992 A071993 KEYWORD easy,nonn AUTHOR Cino Hilliard (hillcino368(AT)gmail.com), Jan 23 2004 STATUS approved

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