A275647 Decimal expansion of Pi^2/6 - Sum_{k>=1} 1/prime(k)^2.

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

Offset: 1

Author

Ilya Gutkovskiy, Aug 04 2016

Keywords

• nonn
• cons

Comments

Decimal expansion of sum of squares of reciprocals of nonprime numbers.

Decimal expansion of the nonprime zeta function at 2.

Continued fraction [1; 5, 5, 3, 1, 2, 2, 6, 2, 2, 4, 1, 1, 93, 2, 1, 1, 5, 3, 5, 3, 2, 1, 2, 6, 1, 4, 5, 1, 34, 1, ...]

More generally, the nonprime zeta function at s equals Sum_{k>=1} (1/k^s - 1/prime(k)^s) = Product_{k>=1} 1/(1 - prime(k)^(-s)) - Sum_{k>=1} 1/prime(k)^s.

Floor(1/(zeta(s)-prime zeta(s)-1)) gives second term in continued fraction for nonprime zeta(s): 5, 36, 187, 852, 3663, 15280, 62692, 254760, 1029279, 4143617, ...

Dirichlet g.f. of A005171: nonprime zeta(s).

Examples

			1/1^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/9^2 + 1/10^2 + ... = 1.192686646807160937965871801813777255045718557966906015999...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Ilya Gutkovskiy, Nonprime zeta function.
• Eric Weisstein's World of Mathematics, Riemann Zeta Function 2.
• Eric Weisstein's World of Mathematics, Prime Zeta Function.

Crossrefs

Cf. A002808, A008683, A013661, A018252, A062312, A085548, A111003, A222171.

Programs

Mathematica

RealDigits[Pi^2/6 - PrimeZetaP[2], 10, 120][[1]]
RealDigits[Zeta[2] - PrimeZetaP[2], 10, 120][[1]]

PARI

eps()=2.>>bitprecision(1.)
primezeta(s)=my(lm=s*log(2)); lm=lambertw(lm/eps())\lm; sum(k=1,lm, moebius(k)*log(abs(zeta(k*s)))/k)
zeta(2) - primezeta(2) \\ Charles R Greathouse IV, Aug 05 2016

PARI

Pi^2/6 - sumeulerrat(1/p, 2) \\ Amiram Eldar, Mar 19 2021

Formula

Equals zeta(2) - prime zeta(2) = A013661 - A085548.

Equals Sum_{k>=1} (1 - k*mu(k)*log(zeta(2*k)))/k^2, where mu(k) is the Moebius function (A008683).

Equals Sum_{k>=1} 1/A062312(k).

Equals Sum_{k>=1} 1/A018252(k)^2.

Equals 1 + Sum_{k>=1} 1/A002808(k)^2.

Equals A222171 + A111003 - A085548.