A356793 Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

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

Offset

0,2

Comments

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

Convergence table:

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

10000000 3285916169 0.165618465394273171950874120818

20000000 7065898967 0.165618465394707600197099741096

30000000 11044807451 0.165618465394836120901019351544

40000000 15151463321 0.165618465394895965582366015390

50000000 19358093939 0.165618465394930089884704869090

60000000 23644223231 0.165618465394951950670948192842

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

Links

Table of n, a(n) for n=0..11.

Jeffrey P.S. Lay, Sign changes in Mertens' first and second theorems, arXiv:1505.03589 [math.NT], 2015.

Mark B. Villarino, Mertens' Proof of Mertens' Theorem, arXiv:math/0504289 [math.HO], 2005.

Marek Wolf, Generalized Brun's constants, IFTUWr 910/97 (1998), 1-15.

Marek Wolf, Some heuristics on the gaps between consecutive primes, arXiv:1102.0481 [math.NT]. 2011.

Example

0.165618465395...

Crossrefs

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

Cf. A347278.

Sequence in context: A185273 A330065 A191220 * A246673 A220190 A231738

Adjacent sequences: A356790 A356791 A356792 * A356794 A356795 A356796

Keyword

nonn,cons,hard,more

Author

Artur Jasinski, Sep 04 2022

Extensions

Data extended to ...3, 9, 5 by Hugo Pfoertner, Sep 28 2022

Status

approved