A306780 Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).

3, 1, 5, 8, 8, 8, 8, 1, 9, 3, 5, 0

Offset: 0

Author

Marco Ripà and Aldo Roberto Pessolano, Mar 09 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 algebraic sum compared with neighboring terms.

Examples

			S_1 - S_2 = 0.315888819350...
		

Crossrefs

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

Programs

PARI

sumalt(j=1, (-1)^(j+1)*(prime(j)!/prime(j + 1)!)) \\ Michel Marcus, Apr 02 2019  \\ Needs default(realprecision, 10^4)  Jinyuan Wang, May 19 2019

Formula

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