A090237 Numerators of the partial sums of the reciprocals of the lower members of twin prime pairs.

1, 8, 103, 1916, 58369, 2474474, 149329111, 10799140436, 1104684106441, 119612262552092, 16537863728067439, 2484826470387072806, 447865969660134667129, 86094083836577627060684

Offset

3,2

Comments

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.

Links

Table of n, a(n) for n=3..16.

Example

1/3+1/5+1/11+1/17 = 1916/2805 and 1916 is the fourth entry in the table.

Mathematica

Accumulate[1/Select[Partition[Prime[Range[50]], 2, 1], #[[2]]-#[[1]]==2&][[All, 1]]]//Numerator (* Harvey P. Dale, Jul 24 2017 *)

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]^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: A333985 A369184 A297069 * A222664 A190786 A138430

Adjacent sequences: A090234 A090235 A090236 * A090238 A090239 A090240

Keyword

easy,nonn

Author

Cino Hilliard, Jan 23 2004

Status

approved