A331369 Decimal expansion of Sum_{(p1, p2) is twin prime pair} 1/p1 + 1/p2 - log(p2/p1).

0, 2, 9, 9, 8, 1, 7, 1, 0, 8, 3, 8, 9, 0, 6, 2, 6, 9, 6, 8, 2, 6

Offset: 0

Author

Dimitris Valianatos, May 03 2020

Keywords

• nonn
• cons
• more

Comments

Let (p_k, p_(k+1)) twin prime pair. Then log(p_(k+1)/p_k) < 1/p_k + 1/p_(k+1).

Lim_{k -> oo} 1/p_k + 1/p_(k+1) - log(p_(k+1)/p_k) = 0.

This constant is analogous to Euler-Mascheroni constant for twin primes.

Examples

			0.0299817108389062696826...
		

Crossrefs

Cf. A077800, A065421, A001359, A006512, A077761.

Programs

PARI

p = 3; st = 0.0; forprime(n = 5, 1e11, if(n - p == 2, st += 1/p + 1/n - log(n/p)); p = n); print(st)

Formula

Equals Sum_{k >= 1} 1/A001359(k) + 1/A006512(k) - log(A006512(k)/A001359(k)).