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

%I A347359 #34 Sep 22 2021 09:41:46
%S A347359 1,2,9,3,3,7,1,7
%N A347359 Decimal expansion of Product_{p in A077800} (1 - 1/p).
%C A347359 Note that A077800 is the sequence of twin primes with 5 repeated. The sequence of twin primes is A001097.
%C A347359 Related to Brun's constant (A065421) and the twin prime constant (A005597).
%C A347359 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.
%C A347359 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).
%C A347359 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.
%D A347359 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.
%H A347359 Ken Hicks and Kevin Ward, <a href="https://arxiv.org/abs/2108.03268">Series and Product Relations Made from Primes</a>, arXiv:2108.03268 [math.NT], 2021.
%e A347359 0.12933717...
%Y A347359 Cf. A001097, A005597, A065421, A077800.
%K A347359 nonn,cons,hard,more
%O A347359 0,2
%A A347359 _Kenneth H. Hicks_, Aug 29 2021
%E A347359 Offset corrected by _N. J. A. Sloane_, Sep 20 2021