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%I #74 Jan 14 2022 07:31:51
%S 7,0,4,4,4,2,2,0,0,9,9,9,1,6,5,5,9,2,7,3,6,6,0,3,3,5,0,3,2,6,6,3,7,2,
%T 1,0,1,8,8,5,8,6,4,3,1,4,1,7,0,9,8,0,4,9,4,1,4,2,2,6,8,4,2,5,9,1,0,9,
%U 7,0,5,6,6,8,2,0,0,6,7,7,8,5,3,6,8,0,8,2,4,4,1,4,5,6,9,3,1,3
%N Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))).
%C The density of A268335. - _Vladimir Shevelev_, Feb 01 2016
%C The probability that two numbers are coprime given that one of them is coprime to a randomly chosen third number. - _Luke Palmer_, Apr 27 2019
%H Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Bordelles/bord14.html">An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions</a>, J. Int. Seq., Vol. 16 (2013), Article 13.6.3.
%H Eckford Cohen, <a href="https://doi.org/10.1007/BF01180473">Arithmetical functions associated with the unitary divisors of an integer</a>, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
%H David Handelman, <a href="http://arxiv.org/abs/1309.7417">Invariants for critical dimension groups and permutation-Hermite equivalence</a>, arXiv preprint arXiv:1309.7417 [math.AC], 2013-2017.
%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arxiv:0903.2514 [math.NT] (2009) constant Q_1^(1).
%H G. Niklasch, <a href="http://guests.mpim-bonn.mpg.de/moree/Moree.en.html">Some number theoretical constants: 1000-digit values</a>.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H V. Sita Ramaiah and D. Suryanarayana, <a href="http://ousar.lib.okayama-u.ac.jp/en/33820">Sums of reciprocals of some multiplicative functions</a>, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
%H R. Sitaramachandrarao and D. Suryanarayana, <a href="https://doi.org/10.1090/S0002-9939-1973-0319922-9">On Sigma_{n<=x} sigma*(n) and Sigma_{n<=x} phi*(n)</a>, Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66.
%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth/toth25.html">Alternating sums concerning multiplicative arithmetic functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1,<a href="https://arxiv.org/abs/1608.00795">arXiv preprint</a>, arXiv:1608.00795 [math.NT], 2016.
%H Deyu Zhang and Wenguang Zhai, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Zhang/zhang10.html">Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n</a>, J. Int. Seq., Vol. 13 (2010), Article 10.4.7, eq. (4).
%H Rimer Zurita <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Zurita/zur3.html">Generalized Alternating Sums of Multiplicative Arithmetic Functions</a>, J. Int. Seq., Vol. 23 (2020), Article 20.10.4.
%F From _Amiram Eldar_, Mar 05 2019: (Start)
%F Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} rad(k), where rad(k) = A007947(k) is the squarefree kernel of k (Cohen).
%F Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} uphi(k), where uphi(k) = A047994(k) is the unitary totient function (Sitaramachandrarao and Suryanarayana).
%F Equals lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/psi(k), where psi(k) = A001615(k) is the Dedekind psi function (Sita Ramaiah and Suryanarayana).
%F (End)
%F Equals A065473*A013661/A065480. - _Luke Palmer_, Apr 27 2019
%F Equals Sum_{k>=1} mu(k)/(k*sigma(k)), where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - _Amiram Eldar_, Jan 14 2022
%e 0.7044422009991655927366033503...
%t $MaxExtraPrecision = 1200; digits = 98; terms = 1200; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Apr 18 2016 *)
%o (PARI) prodeulerrat(1 - 1/(p*(p+1))) \\ _Amiram Eldar_, Mar 14 2021
%Y Cf. A001615, A007947, A047994, A078082, A268335, A306633.
%Y Cf. A000203, A065473, A065480, A065490, A008683.
%K cons,nonn
%O 0,1
%A _N. J. A. Sloane_, Nov 19 2001
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