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

0,2

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.

Links

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

Wikipedia, Rosser's theorem

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) +...

Example

S_1* = 0.085161910985...

Crossrefs

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

Sequence in context: A334496 A258988 A139721 * A261882 A021058 A036792

Adjacent sequences: A307381 A307382 A307383 * A307385 A307386 A307387

Keyword

cons,nonn,more

Author

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

Status

approved