A307384 -id:A307384 - cp's OEIS frontend

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

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

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Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

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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.

			S_2* = 0.045294348850...

Wikipedia, Rosser's theorem

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

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) +...

cp's OEIS Frontend

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

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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