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A136141 Decimal expansion of Sum_{p prime} 1/(p*(p-1)).

%I A136141 #78 Jul 01 2025 09:27:58
%S A136141 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,
%T A136141 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,
%U A136141 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
%N A136141 Decimal expansion of Sum_{p prime} 1/(p*(p-1)).
%C A136141 Excess of prime factors with multiplicity over distinct prime factors for random (large) integers. - _Charles R Greathouse IV_, Sep 06 2011
%C A136141 Sum of reciprocals of (proper) prime powers. The sum of reciprocals of all proper powers is A072102. - _Charles R Greathouse IV_, Apr 24 2012
%C A136141 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
%C A136141 See the second 'Applications' example under the Mathematica help file for the function PrimePowerQ. - _Robert G. Wilson v_, May 13 2019
%C A136141 It easy to prove that this constant < 1 (Sum_{p prime} 1/(p*(p-1)) < Sum_{k>=2} 1/(k*(k-1)) = 1). Luthar (1969) asks for a better upper bound. The solution shows that this constant is < 3/2 - log(2) = 0.80685... . - _Amiram Eldar_, Feb 14 2025
%D A136141 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D A136141 Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants, p. 94.
%H A136141 Robert G. Wilson v, <a href="/A136141/b136141.txt">Table of n, a(n) for n = 0..10000</a> (first 1002 terms from Jason Kimberley).
%H A136141 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision computation of Hardy-Littlewood constants</a>, preprint 1991.
%H A136141 Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H A136141 Stephan Ramon Garcia and Ethan Simpson Lee, <a href="https://arxiv.org/abs/2506.17416">Explicit conditional bounds for the residue of a Dedekind zeta-function at s = 1</a>, arXiv:2506.17416 [math.NT], 2025. See p. 3.
%H A136141 R. S. Luthar, <a href="http://www.jstor.org/stable/2317957">Problem E 2192</a>, Elementary Problems, The American Mathematical Monthly, Vol. 76, No. 8 (1969), p. 938; <a href="http://www.jstor.org/stable/2316224">An Upper Bound</a>, Solution to Problem E 2192, by O. P. Lossers, ibid., Vol. 77, No. 7 (1970), pp. 769-770.
%H A136141 R. J. Mathar, <a href="http://arxiv.org/abs/0803.0900">Series of reciprocal powers of k-almost primes</a>, arXiv:0803.0900 [math.NT], 2008-2009. Tables 8 and 10.
%H A136141 Michael Ian Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/PST.pdf">Property Enumerators and a Partial Sum Theorem</a>, 2011; <a href="https://citeseerx.ist.psu.edu/pdf/e503bec3c04c3f94cb267882724dd414e143141b">alternative link</a>.
%H A136141 <a href="/wiki/Index_to_constants#Start_of_section_P">Index to constants which are prime zeta sums</a> {1,1,0}
%F A136141 Equals Sum_{n>=1} 1/A036689(n).
%F A136141 Equals Sum_{s>=2} P(s), where P is the prime zeta function. - _Charles R Greathouse IV_, Sep 06 2011
%F A136141 Equals A083342 - A077761, that is, Sum_{n>=2} ((EulerPhi(n) - MoebiusMu(n))/n) * log(zeta(n)). - _Jean-François Alcover_, Sep 02 2015
%F A136141 Equals 2 * Sum_{k>=2} pi(k)/(k^3-k), where pi(k) = A000720(k) (Shamos, 2011, p. 8). - _Amiram Eldar_, Mar 12 2024
%e A136141 Equals 1/2 + 1/(3*2) + 1/(5*4) + 1/(7*6) + ...
%e A136141 = 0.7731566690497951278643674598559423956187413360831860483110060673567...
%t A136141 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 *)
%o A136141 (PARI) W(x)=solve(y=log(x)/2,max(1,log(x)),y*exp(y)-x)
%o A136141 eps()=2. >> (32*ceil(default(realprecision)/9.63))
%o A136141 primezeta(s)=my(t=s*log(2),iter=W(t/eps())\t);sum(k=1,iter, moebius(k)/k*log(abs(zeta(k*s))))
%o A136141 a(lim,e)={ \\ choose parameters to maximize speed and precision
%o A136141     my(x,y=exp(W(lim)-.5));
%o A136141     x=lim^e*(e*log(y))^e*(y*log(y))^-e*incgam(-e,e*log(y));
%o A136141     forprime(p=2,lim,x+=1/((p*1.)^e*(p-1)));
%o A136141     x+sum(n=2,e,primezeta(n))
%o A136141 }; \\ _Charles R Greathouse IV_, Sep 07 2011
%o A136141 (PARI) sumeulerrat(1/(p*(p-1))) \\ _Amiram Eldar_, Mar 18 2021
%o A136141 (Magma) R := RealField(105);
%o A136141 c := &+[R|(EulerPhi(n)-MoebiusMu(n))/n*Log(ZetaFunction(R,n)):n in[2..360]];
%o A136141 Reverse(IntegerToSequence(Floor(c*10^103))); // _Jason Kimberley_, Jan 12 2017
%Y A136141 Cf. A152447 (over the semiprimes), A000040, A000720, A001248, A046660 (excess, see first comment), A072102, A077761, A083342, A179119, A246547.
%Y A136141 Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
%K A136141 cons,easy,nonn
%O A136141 0,1
%A A136141 _R. J. Mathar_, Mar 09 2008
%E A136141 More terms from _D. S. McNeil_, Sep 06 2011
%E A136141 More digits from _Jean-François Alcover_, Sep 02 2015