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A136141
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Decimal expansion of Sum_{p prime} 1/(p*(p-1)).
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35
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7, 7, 3, 1, 5, 6, 6, 6, 9, 0, 4, 9, 7, 9, 5, 1, 2, 7, 8, 6, 4, 3, 6, 7, 4, 5, 9, 8, 5, 5, 9, 4, 2, 3, 9, 5, 6, 1, 8, 7, 4, 1, 3, 3, 6, 0, 8, 3, 1, 8, 6, 0, 4, 8, 3, 1, 1, 0, 0, 6, 0, 6, 7, 3, 5, 6, 7, 0, 9, 0, 2, 8, 4, 8, 9, 2, 3, 3, 3, 9, 7, 8, 3, 3, 7, 9, 8, 7, 5, 8, 8, 2, 3, 3, 2, 0, 8, 1, 8, 3, 2, 8, 9
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OFFSET
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0,1
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COMMENTS
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Excess of prime factors with multiplicity over distinct prime factors for random (large) integers. - Charles R Greathouse IV, Sep 06 2011
Decimal expansion of the infinite sum of the reciprocals of the prime powers which are not prime (A246547). - Robert G. Wilson v, May 13 2019
See the second 'Applications' example under the Mathematica help file for the function PrimePowerQ. - Robert G. Wilson v, May 13 2019
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REFERENCES
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Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants, p. 94.
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LINKS
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FORMULA
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Equals 2 * Sum_{k>=2} pi(k)/(k^3-k), where pi(k) = A000720(k) (Shamos, 2011, p. 8). - Amiram Eldar, Mar 12 2024
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EXAMPLE
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Equals 1/2 + 1/(3*2) + 1/(5*4) + 1/(7*6) + ...
= 0.7731566690497951278643674598559423956187413360831860483110060673567...
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MATHEMATICA
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digits = 103; sp = NSum[PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)
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PROG
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(PARI) W(x)=solve(y=log(x)/2, max(1, log(x)), y*exp(y)-x)
eps()=2. >> (32*ceil(default(realprecision)/9.63))
primezeta(s)=my(t=s*log(2), iter=W(t/eps())\t); sum(k=1, iter, moebius(k)/k*log(abs(zeta(k*s))))
a(lim, e)={ \\ choose parameters to maximize speed and precision
my(x, y=exp(W(lim)-.5));
x=lim^e*(e*log(y))^e*(y*log(y))^-e*incgam(-e, e*log(y));
forprime(p=2, lim, x+=1/((p*1.)^e*(p-1)));
x+sum(n=2, e, primezeta(n))
(PARI) sumeulerrat(1/(p*(p-1))) \\ Amiram Eldar, Mar 18 2021
(Magma) R := RealField(105);
c := &+[R|(EulerPhi(n)-MoebiusMu(n))/n*Log(ZetaFunction(R, n)):n in[2..360]];
Reverse(IntegerToSequence(Floor(c*10^103))); // Jason Kimberley, Jan 12 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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