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

Author

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

Keywords

• cons
• nonn
• more

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.

Examples

			S_1* + S_2* = 0.130456269835...
		

Crossrefs

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

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) + ...