A306744 Decimal expansion of the constant S_1 + S_2 = Sum_{j>=1} prime(j)!/prime(j + 1)!.

4, 1, 9, 2, 2, 2, 0, 6, 4, 9, 0, 3

Offset: 0

Author

Marco Ripà, Mar 07 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.

Examples

			S_1 + S_2 = 0.419222064903...
		

Crossrefs

Cf. A000040, A306658 (S_1), A306700 (S_2), A306780.

Programs

Mathematica

s = 0; p = 2; q = 3; While[p < 10^10, s = N[s + 1/Times @@ Range[p +1, q], 32]; p = q; q = NextPrime@ q]; Take[ RealDigits[s][[1]], 20] (* Robert G. Wilson v, Mar 23 2019 *)

PARI

suminf(j=1, prime(j)!/prime(j + 1)!) \\ Michel Marcus, Apr 02 2019

Formula

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