A307384 Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2*j) - 1)!/prime((2*j + 1) - 1)!.

0, 8, 5, 1, 6, 1, 9, 1, 0, 9, 8, 5

Offset: 0

Author

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

Keywords

• cons
• nonn
• more

Comments

Together with the constant S_2* and S_1* + S_2* (see A307383), S_1* involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.

On Apr 06 2019, the first 4200000000 prime numbers were used and using Rosser's theorem we get: 0.08516191098523 < S_1* < 0.08516191098543.

Examples

			S_1* = 0.085161910985...
		

Links

• Wikipedia, Rosser's theorem

Crossrefs

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

Formula

S_1* = Sum_{j>=1} prime(2*j - 1)!/prime((2*j + 1) - 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j), prime(2*j + 1)} k) = 1/(3*4) + 1/(7*8*9*10) + 1/(13*14*15*16) + 1/(19*20*21*22) +...