A307383 Decimal expansion of the constant S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)!.

1, 3, 0, 4, 5, 6, 2, 5, 9, 8, 3, 5

Offset

0,2

Comments

The constant S_1* + 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: 0.13045626983537 < S_1* + S_2* < 0.13045626983578.

Links

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

Formula

S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)! = Sum_{j>=1} 1/(Product{k=prime(j + 1), prime((j + 2) - 1)} k) = 1/(3*4) + 1/(5*6) + 1/(7*8*9*10) + 1/(11*12) + ...

Example

S_1* + S_2* = 0.130456269835...

Crossrefs

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

Sequence in context: A137335 A011077 A322936 * A111486 A192878 A126826

Adjacent sequences: A307380 A307381 A307382 * A307384 A307385 A307386

Keyword

cons,nonn,more

Author

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

Status

approved