This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A209329 #41 Nov 24 2021 00:40:18
%S A209329 1,3,4,4,2,6,5,0,9,6,9,1,7,3,3,2,2,8
%N A209329 Decimal expansion of the sum over the inverse products of adjacent odd primes.
%C A209329 Contains the contribution from twin primes (A209328) plus other contributions from cousin primes (A143206) not already part of twin primes, sexy primes (A210477) not already accounted for, etc.
%C A209329 Summing up to (and including) 12-digit primes yields 0.134426509691698261. - _Hans Havermann_, Mar 17 2013
%H A209329 R. J. Mathar, <a href="/A209329/a209329.pdf">Estimate of the Sum over Inverse Products of Adjacent Prime Pairs</a>
%F A209329 sum_{3 < p < 10^4} 1/(prevprime(p)*p) = 0.134416688[9]...
%F A209329 sum_{3 < p < 10^5} 1/(prevprime(p)*p) = 0.134425707...
%F A209329 sum_{3 < p < 10^6} 1/(prevprime(p)*p) = 0.1344264419...
%F A209329 sum_{3 < p < 10^7} 1/(prevprime(p)*p) = 0.13442650383...
%F A209329 sum_{3 < p < 10^8} 1/(prevprime(p)*p) = 0.13442650917[5]...
%F A209329 sum_{3 < p < 10^9} 1/(prevprime(p)*p) = 0.13442650964545...
%F A209329 Extrapolation of this data (using Aitken's method) indeed suggests a value of 0.134426509692, rounded to the last decimal place. Extrapolation of the ratios of the first differences (9.02e-6, 7.35e-7, 6.19e-8, 5.34e-9, 4.699e-10) yields subsequent terms (4.26e-11, 4.0e-12). - _M. F. Hasler_, Jan 22 2013
%e A209329 0.134426509... = 1/(3*5) + 1/(5*7) + 1/(7*11) + 1/(11*13)+ ... = Sum_{n>=2} 1/A006094(n).
%o A209329 (PARI) {default(realprecision,19);s=0;q=1/3;forprime(p=1/q+1,10^9,s+=q*q=1./p);s} /* _M. F. Hasler_, Jan 22 2013 */
%Y A209329 Cf. A210473 (includes 1/(2*3)). Cf. also A085548.
%K A209329 cons,nonn,more
%O A209329 0,2
%A A209329 _R. J. Mathar_, Jan 19 2013
%E A209329 More terms from _R. J. Mathar_, Feb 08 2013