A307385 Decimal expansion of the constant S_2* = Sum_{j>=1} prime((2*j + 1) - 1)!/prime((2*j + 2) - 1)!.

0, 4, 5, 2, 9, 4, 3, 4, 8, 8, 5, 0

Offset

0,2

Comments

The constant S_2* is related to the prime gaps, since twin primes produce the largest terms of the sum compared with neighboring terms.

On Apr 06 2019, the first 4200000000 prime numbers were used in order to calculate S_1* and S_2* and using Rosser's theorem we get: 0.04529434885014 < S_1* + S_2* < 0.04529434885035.

Links

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

Wikipedia, Rosser's theorem

Formula

S_2* = Sum_{j>=1} prime((2*j + 1) - 1)!/prime((2*j + 2) - 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j + 1), prime((2*j + 2) - 1)} k) = 1/(5*6) + 1/(11*12) + 1/(17*18) + 1/(23*24*25*26*27*28) +...

Example

S_2* = 0.045294348850...

Crossrefs

Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2), A307383 (S_1* + S_2*), A307384 (S_1*).

Sequence in context: A281143 A200684 A345210 * A354700 A046572 A046574

Adjacent sequences: A307382 A307383 A307384 * A307386 A307387 A307388

Keyword

cons,nonn,more

Author

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

Status

approved