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%I #32 Mar 18 2021 07:33:15
%S 8,0,7,3,3,0,8,2,1,6,3,6,2,0,5,0,3,9,1,4,8,6,5,4,2,7,9,9,3,0,0,3,1,1,
%T 3,4,0,2,5,8,4,5,8,2,5,0,8,1,5,5,6,6,4,4,0,1,8,0,0,5,2,0,7,7,0,4,4,1,
%U 3,8,1,4,8,4,9,3,7,5,1,8,6,4,9,6,9,5,6,0,9,3,5,0,9,6,2,9,4,8,3,7,6,5,0,1,1,8
%N Decimal expansion of Sum_{n>=1} (-1)^omega(n) phi(n)^2/n^4, where omega(n) is the number of distinct prime factors of n (A001221) and phi is Euler's totient function (A000010).
%C The constant A that appears in the asymptotic formulae for the sums of the bi-unitary divisor function (A306069) and the bi-unitary totient function (A306070).
%C The product in Suryanarayana's 1972 paper has a error that was corrected in his 1975 paper.
%C The probability that 2 randomly selected numbers will be unitary coprime (i.e. their largest common unitary divisor is 1). - _Amiram Eldar_, Aug 27 2019
%D József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.
%H D. Suryanarayana, <a href="https://doi.org/10.1007/BFb0058797">The number of bi-unitary divisors of an integer</a>, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
%H D. Suryanarayana and R. Sita Rama Chandra Rao, <a href="http://informaticsjournals.com/index.php/jims/article/view/16651">The number of bi-unitary divisors of an integer - II</a>, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.
%H László Tóth, <a href="http://emis.ams.org/journals/JIS/VOL12/Toth2/toth5.html">On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.
%H László Tóth, <a href="https://doi.org/10.1007/978-1-4939-1106-6_19">Multiplicative arithmetic functions of several variables: a survey</a>, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 508), <a href="https://arxiv.org/abs/1310.7053">preprint</a>, arXiv:1310.7053 [math.NT] (2014) (see p. 21).
%F Equals Product_{p prime} (1 - (p-1)/(p^2 * (p+1))).
%F Equals zeta(2) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4).
%e 0.80733082163620503914...
%t cc = CoefficientList[Series[Log[1 - (p - 1)/(p^2*(p + 1))] /. p -> 1/x, {x, 0, 36}], x]; f = FindSequenceFunction[cc]; digits = 20; A = Exp[NSum[f[n + 1 // Floor]*(PrimeZetaP[n]), {n, 2, Infinity}, NSumTerms -> 16 digits, WorkingPrecision -> 16 digits]]; RealDigits[A, 10, digits][[1]] (* _Jean-François Alcover_, Jun 19 2018 *)
%t $MaxExtraPrecision = 1000; Do[Print[Zeta[2] * Exp[-N[Sum[q = Expand[(2*p^2 - 2*p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}], 120]]], {t, 300, 1000, 100}] (* _Vaclav Kotesovec_, May 29 2020 *)
%o (PARI) prodeulerrat(1 - (p-1)/(p^2 * (p+1))) \\ _Amiram Eldar_, Mar 18 2021
%Y Cf. A000010, A001221, A286324, A306069, A306070.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Jun 19 2018
%E a(1)-a(20) from _Jean-François Alcover_, Jun 19 2018
%E a(20)-a(24) from _Jon E. Schoenfield_, May 27 2019
%E More terms from _Vaclav Kotesovec_, May 29 2020
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