cp's OEIS Frontend

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

%I A307379 #30 Dec 05 2024 18:58:20
%S A307379 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,
%T A307379 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,
%U A307379 5,5,6,6,6,4,1,4,1,8,4,1,1,1,3,5,6,5,9
%N A307379 Decimal expansion of Sum_{n >= 1} 2/(k(n)*(k(n) + 1)), with k = A018252 (nonprime numbers).
%C A307379 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).
%F A307379 Equals 2*(1 - A179119) = 2*(1 - Sum_{n>=1} 1/(A000040(n)*A008864(n))).
%e A307379 1.3395401474715935179... = 2 - (1/3 + 1/(3*2) + 1/(5*3) + 1/(7*4) + 1/(11*6) + ...) = 2*(1 - A179119).
%t A307379 digits = 87;
%t A307379 S = 2 - 2 NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5];
%t A307379 RealDigits[S, 10, digits][[1]] (* _Jean-François Alcover_, Jun 20 2019 *) [From A179119]
%o A307379 (PARI) 2*(1 - sumeulerrat(1/(p*(p+1)))) \\ _Amiram Eldar_, Mar 18 2021
%Y A307379 Cf. A000040, A008864, A018252, A179119.
%K A307379 cons,easy,nonn
%O A307379 1,2
%A A307379 _Marco Ripà_, Apr 06 2019
%E A307379 Edited by _Wolfdieter Lang_, Jul 10 2019