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Decimal expansion of Sum_{p prime} 1/(p^3 - 1).
3

%I #10 Mar 18 2021 07:33:44

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

%T 3,7,4,5,1,8,7,7,0,1,8,4,5,2,7,9,2,1,9,7,7,3,5,0,4,3,4,9,4,0,4,1,0,3,

%U 8,0,8,7,4,2,0,5,7,9,2,5,2,6,3,3,9,3,9,5,3,9,8,7,7,6,5,4,3,5,3,6,7,8,8,2,3

%N Decimal expansion of Sum_{p prime} 1/(p^3 - 1).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>.

%F Equals Sum_{k>=1} primezeta(3*k).

%F More generally, Sum_{p prime} 1/(p^s - 1) = Sum_{k>=1} primezeta(s*k).

%e 1/(2^3 - 1) + 1/(3^3 - 1) + 1/(5^3 - 1) + ... = 1/2^3 + 1/3^3 + 1/5^3 + ... + 1/2^6 + 1/3^6 + 1/5^6 + ... + 1/2^9 + 1/3^9 + 1/5^9 + ... = 0.19411816983263379229...

%t digits = 105; sp = NSum[PrimeZetaP[3 n], {n, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First

%o (PARI) sumeulerrat(1/(p^3-1)) \\ _Amiram Eldar_, Mar 18 2021

%Y Cf. A030078, A085541, A154945.

%K nonn,cons

%O 0,2

%A _Ilya Gutkovskiy_, May 04 2017