A356793 - OEIS

%I #65 Sep 29 2022 22:05:29

%S 1,6,5,6,1,8,4,6,5,3,9,5

%N Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

%C Alternative definition: sum of squares of reciprocals of primes whose distance from the next prime is equal to 2.

%C Convergence table:

%C    k       A001359(k)      Sum_{j=1..k} 1/A001359(j)^2

%C 10000000   3285916169 0.165618465394273171950874120818

%C 20000000   7065898967 0.165618465394707600197099741096

%C 30000000  11044807451 0.165618465394836120901019351544

%C 40000000  15151463321 0.165618465394895965582366015390

%C 50000000  19358093939 0.165618465394930089884704869090

%C 60000000  23644223231 0.165618465394951950670948192842

%C Using the Hardy-Littlewood prediction of the density of twin primes (see A347278), the contribution to the sum after the last entry in the table above can be estimated as 9.056*10^(-14), making the infinite sum ~= 0.16561846539504... . - _Hugo Pfoertner_, Sep 28 2022

%H Jeffrey P.S. Lay, <a href="https://arxiv.org/abs/1505.03589">Sign changes in Mertens' first and second theorems</a>, arXiv:1505.03589 [math.NT], 2015.

%H Mark B. Villarino, <a href="https://arxiv.org/abs/math/0504289">Mertens' Proof of Mertens' Theorem</a>, arXiv:math/0504289 [math.HO], 2005.

%H Marek Wolf, <a href="https://www.researchgate.net/publication/2346256_Generalized_Brun%27s_constants">Generalized Brun's constants</a>, IFTUWr 910/97 (1998), 1-15.

%H Marek Wolf, <a href="https://arxiv.org/abs/1102.0481">Some heuristics on the gaps between consecutive primes</a>, arXiv:1102.0481 [math.NT]. 2011.

%e 0.165618465395...

%Y Cf. A006512, A065421, A077800, A078437, A085548, A096247, A160910, A194098, A209328, A209329, A242301, A242302, A242303, A242304, A306539, A342714.

%Y Cf. A347278.

%K nonn,cons,hard,more

%O 0,2

%A _Artur Jasinski_, Sep 04 2022

%E Data extended to ...3, 9, 5 by _Hugo Pfoertner_, Sep 28 2022