A347359 Decimal expansion of Product_{p in A077800} (1 - 1/p).

1, 2, 9, 3, 3, 7, 1, 7

Offset

0,2

Comments

Note that A077800 is the sequence of twin primes with 5 repeated. The sequence of twin primes is A001097.

Related to Brun's constant (A065421) and the twin prime constant (A005597).

It is well known that the product of 1-1/p over all primes p is zero (it is related to the Riemann zeta function). Also the sum of 1/p diverges, whereas the sum of 1/p2 for p2 in the sequence A077800 converges to Brun's constant, regardless of whether there are an infinite number of twin primes or not. Similarly, the product in the present sequence also converges.

The repeated value of 1/5 is used in the calculation of Brun's constant (A065421) and we follow that convention here. The first two pairs of twin primes are (3,5) and (5,7), so the 4 initial terms in the product are (1-1/3)*(1-1/5)*(1-1/5)*(1-1/7).

This constant converges very slowly, similar to the convergence of Brun's constant. For example, for all twin primes below 1 billion, the product only reaches the value of 0.1469... Details on the error term in the convergence of the above product will be given in a forthcoming paper.

References

K. Hicks and K. Ward, Series and Product Relations Made from Primes, Pi Mu Epsilon Journal, Vol. 15, No. 3, Fall 2020, pp. 161-169.

Links

Table of n, a(n) for n=0..7.

Ken Hicks and Kevin Ward, Series and Product Relations Made from Primes, arXiv:2108.03268 [math.NT], 2021.

Example

0.12933717...

Crossrefs

Cf. A001097, A005597, A065421, A077800.

Sequence in context: A021345 A011066 A071194 * A346839 A137616 A154581

Adjacent sequences: A347356 A347357 A347358 * A347360 A347361 A347362

Keyword

nonn,cons,hard,more

Author

Kenneth H. Hicks, Aug 29 2021

Extensions

Offset corrected by N. J. A. Sloane, Sep 20 2021

Status

approved