A071990 Numerators of the partial sums of the reciprocals of the upper members of twin prime pairs.

1, 12, 191, 4084, 135249, 6083702, 382629607, 28634912196, 3000711370793, 332363027120752, 46774578929554863, 7143041842570860878, 1304982717560745380233, 254050342563254798982984, 50978433038375676533030711, 11758121724577336261611994924

Offset

1,2

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

Table of n, a(n) for n=1..16.

Formula

a(n) = numerator(Sum_{k=1..n} 1 / A006512(k)). - Sean A. Irvine, Aug 25 2024

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

Cf. A006512.

Sequence in context: A239294 A294409 A239776 * A230757 A196483 A196716

Adjacent sequences: A071987 A071988 A071989 * A071991 A071992 A071993

Keyword

easy,nonn

Author

Cino Hilliard, Jan 23 2004

Extensions

Offset corrected and more terms from Sean A. Irvine, Aug 25 2024

Status

approved