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 A065463 Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))). 24
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The density of A268335. - Vladimir Shevelev, Feb 01 2016 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 LINKS Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013), Article 13.6.3. Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80. Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50. David Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013-2017. R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arxiv:0903.2514 [math.NT] (2009) constant Q_1^(1). G. Niklasch, Some number theoretical constants: 1000-digit values. G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy] V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164. R. Sitaramachandrarao and D. Suryanarayana, On Sigma_{n<=x} sigma*(n) and Sigma_{n<=x} phi*(n), Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66. László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1,arXiv preprint, arXiv:1608.00795 [math.NT], 2016. Deyu Zhang and Wenguang Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq., Vol. 13 (2010), Article 10.4.7, eq. (4). Rimer Zurita Generalized Alternating Sums of Multiplicative Arithmetic Functions, J. Int. Seq., Vol. 23 (2020), Article 20.10.4. FORMULA From Amiram Eldar, Mar 05 2019: (Start) 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). 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). 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). (End) Equals A065473*A013661/A065480. - Luke Palmer, Apr 27 2019 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 EXAMPLE 0.7044422009991655927366033503... MATHEMATICA \$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 *) PROG (PARI) prodeulerrat(1 - 1/(p*(p+1))) \\ Amiram Eldar, Mar 14 2021 CROSSREFS Cf. A001615, A007947, A047994, A078082, A268335, A306633. Cf. A000203, A065473, A065480, A065490, A008683. Sequence in context: A021146 A201424 A070513 * A319739 A242780 A324997 Adjacent sequences:  A065460 A065461 A065462 * A065464 A065465 A065466 KEYWORD cons,nonn AUTHOR N. J. A. Sloane, Nov 19 2001 STATUS approved

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Last modified July 2 14:28 EDT 2022. Contains 355007 sequences. (Running on oeis4.)