A307379 Decimal expansion of Sum_{n >= 1} 2/(k(n)*(k(n) + 1)), with k = A018252 (nonprime numbers).

1, 3, 3, 9, 5, 4, 0, 1, 4, 7, 4, 7, 1, 5, 9, 3, 5, 1, 7, 9, 6, 9, 8, 1, 0, 8, 2, 3, 8, 2, 6, 5, 1, 0, 4, 7, 8, 7, 1, 1, 4, 8, 1, 1, 6, 1, 0, 5, 1, 8, 5, 9, 0, 8, 7, 6, 9, 9, 5, 4, 2, 7, 9, 8, 4, 7, 5, 1, 5, 5, 6, 6, 6, 4, 1, 4, 1, 8, 4, 1, 1, 1, 3, 5, 6, 5, 9

Offset: 1

Author

Marco Ripà, Apr 06 2019

Keywords

• cons
• easy
• nonn

Comments

We know that Sum_{n >= 1} 2/(n^2 + n) = 2 and Sum_{n >= 1} 2/(p(n)*(p(n) + 1)) = 2*A179119, where p = A000040. Therefore, the present decimal expansion 1/1 + 1/10 + 1/21 + 1/36 + ... = 2*(1 - A179119).

Examples

			1.3395401474715935179... = 2 - (1/3 + 1/(3*2) + 1/(5*3) + 1/(7*4) + 1/(11*6) + ...) = 2*(1 - A179119).
		

Crossrefs

Cf. A000040, A008864, A018252, A179119.

Programs

Mathematica

digits = 87;
S = 2 - 2 NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5];
RealDigits[S, 10, digits][[1]] (* Jean-François Alcover, Jun 20 2019 *) [From A179119]

PARI

2*(1 - sumeulerrat(1/(p*(p+1)))) \\ Amiram Eldar, Mar 18 2021

Formula

Equals 2*(1 - A179119) = 2*(1 - Sum_{n>=1} 1/(A000040(n)*A008864(n))).

Extensions

Edited by Wolfdieter Lang, Jul 10 2019