A209329 Decimal expansion of the sum over the inverse products of adjacent odd primes.

1, 3, 4, 4, 2, 6, 5, 0, 9, 6, 9, 1, 7, 3, 3, 2, 2, 8

Offset: 0

Author

R. J. Mathar, Jan 19 2013

Keywords

• cons
• nonn
• more

Comments

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.

Summing up to (and including) 12-digit primes yields 0.134426509691698261. - Hans Havermann, Mar 17 2013

Examples

			0.134426509... = 1/(3*5) + 1/(5*7) + 1/(7*11) + 1/(11*13)+ ... = Sum_{n>=2} 1/A006094(n).
		

Links

• R. J. Mathar, Estimate of the Sum over Inverse Products of Adjacent Prime Pairs

Crossrefs

Cf. A210473 (includes 1/(2*3)). Cf. also A085548.

Programs

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 */

Formula

sum_{3 < p < 10^4} 1/(prevprime(p)*p) = 0.134416688[9]...

sum_{3 < p < 10^5} 1/(prevprime(p)*p) = 0.134425707...

sum_{3 < p < 10^6} 1/(prevprime(p)*p) = 0.1344264419...

sum_{3 < p < 10^7} 1/(prevprime(p)*p) = 0.13442650383...

sum_{3 < p < 10^8} 1/(prevprime(p)*p) = 0.13442650917[5]...

sum_{3 < p < 10^9} 1/(prevprime(p)*p) = 0.13442650964545...

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

Extensions

More terms from R. J. Mathar, Feb 08 2013