login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime>=2} 1/p^2. 33
4, 5, 2, 2, 4, 7, 4, 2, 0, 0, 4, 1, 0, 6, 5, 4, 9, 8, 5, 0, 6, 5, 4, 3, 3, 6, 4, 8, 3, 2, 2, 4, 7, 9, 3, 4, 1, 7, 3, 2, 3, 1, 3, 4, 3, 2, 3, 9, 8, 9, 2, 4, 2, 1, 7, 3, 6, 4, 1, 8, 9, 3, 0, 3, 5, 1, 1, 6, 5, 0, 2, 7, 3, 6, 3, 9, 1, 0, 8, 7, 4, 4, 4, 8, 9, 5, 7, 5, 4, 4, 3, 5, 4, 9, 0, 6, 8, 5, 8, 2, 2, 2, 8, 0, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98

LINKS

Table of n, a(n) for n=0..104.

H. Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint.

Persi Diaconis, Frederick Mosteller, Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors-including the square free case, J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).

X. Gourdon and P. Sebah, Some Constants from Number theory

Gerhard Niklasch and Pieter Moree, Some number-theoretical constants [Cached copy]

Eric Weisstein's World of Mathematics, Prime Sums

Eric Weisstein's World of Mathematics, Prime Zeta Function

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Wikipedia, Prime Zeta Function

FORMULA

P(2) = Sum_{p prime>=2} 1/p^2 = Sum_{n=1..inf} mobius(n)*log(zeta(2*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Equals A085991 + A086032 + 1/4. - R. J. Mathar, Jul 22 2010

EXAMPLE

0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2+...

MAPLE

A085548:= proc(i) print(evalf(add(1/ithprime(k)^2, k=1..i), 100)); end:

A085548(100000); Paolo P. Lava, May 29 2012

MATHEMATICA

If [$VersionNumber < 7.0, m = 200; $MaxExtraPrecision = 200; PrimeZetaP[s_] := NSum[MoebiusMu[k]*Log[Zeta[k*s]]/k, {k, 1, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]]; RealDigits[PrimeZetaP[2]][[1]][[1 ;; 105]] (* Jean-Fran├žois Alcover, Jun 24 2011 *)

PROG

(PARI) recip2(n) = { v=0; p=1; forprime(y=2, n, v=v+1./y^2; ); print(v) }

(PARI) eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)), 9)

lm=lambertw(log(4)/eps())\log(4);

sum(k=1, lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ Charles R Greathouse IV, Jul 19 2013

CROSSREFS

Cf. A085541 (at 3), A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.

Cf. A013661, A078437, A242301.

Sequence in context: A267095 A016715 A255701 * A074459 A155793 A070593

Adjacent sequences:  A085545 A085546 A085547 * A085549 A085550 A085551

KEYWORD

easy,nonn,cons

AUTHOR

Cino Hilliard, Jul 03 2003

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Offset corrected by R. J. Mathar, Feb 05 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 10 23:13 EST 2016. Contains 279021 sequences.