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Search: id:A090237
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| A090237 |
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Numerators of the partial sums of the reciprocals of the lower members of twin prime pairs. |
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+0
1
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| 1, 8, 103, 1916, 58369, 2474474, 149329111, 10799140436, 1104684106441, 119612262552092, 16537863728067439, 2484826470387072806, 447865969660134667129, 86094083836577627060684
(list; graph; listen)
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OFFSET
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3,2
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COMMENT
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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.
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EXAMPLE
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1/3+1/5+1/11+1/17 = 1916/2805 and 1916 is the fourth entry in the table.
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PROGRAM
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(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++; ) ) }
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CROSSREFS
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Sequence in context: A083227 A060570 A001575 this_sequence A138430 A109774 A001657
Adjacent sequences: A090234 A090235 A090236 this_sequence A090238 A090239 A090240
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KEYWORD
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easy,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Jan 23 2004
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