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%I #33 Sep 09 2023 06:51:43
%S 2,1,7,7,7,8,7,1,6,6,1,9,5,3,6,3,7,8,3,2,3,0,0,7,5,1,4,1,1,9,4,4,6,8,
%T 1,3,1,3,0,7,9,7,7,5,5,0,0,1,3,5,5,9,3,7,6,4,8,2,7,6,4,0,3,5,2,3,6,2,
%U 6,4,9,1,1,1,2,2,5,2,6,2,0,5,5,7,9,2,5,4,4,3,8,2,3,5,6,3,7,6,5,6,9,1,8,3,3,9
%N Decimal expansion of Product_{p prime} (1-4/p^2+4/p^3-1/p^4).
%C Also decimal expansion of the probability that an integer tuple (x,y,z,w) satisfies gcd(x,y) = gcd(y,z) = gcd(z,w) = gcd(w,x) = 1.
%H Juan Arias-de-Reyna, <a href="/A256392/b256392.txt">Table of n, a(n) for n = 0..1000</a>
%H Juan Arias de Reyna and Randell Heyman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Heyman/heyman6.html">Counting Tuples Restricted by Pairwise Coprimality Conditions</a>, J. Int. Seq., Vol. 18 (2015), Article 15.10.4; <a href="http://arxiv.org/abs/1403.2769">arXiv preprint</a>, arXiv:1403.2769 [math.NT], 2014.
%e 0.2177787166195363783230075141...
%t Do[Print[N[Exp[-Sum[q = Expand[(4 p^2 - 4 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}]], 50]], {t, 10, 100, 10}] (* _Vaclav Kotesovec_, Dec 17 2019 *)
%o (PARI) prodeulerrat(1-4/p^2+4/p^3-1/p^4) \\ _Amiram Eldar_, Mar 03 2021
%Y Cf. A065473, A196524, A256391, A330523, A336643.
%K nonn,cons
%O 0,1
%A _Juan Arias-de-Reyna_, Mar 28 2015
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