This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A273093 #25 Feb 16 2025 08:33:35
%S A273093 1,7,4,2,1,9,7,8,3,0,3,4,7,2,4,7,0,0,5,5,8,5,7,4,0,7,2,1,8,0,5,3,4,6,
%T A273093 9,1,6,5,1,1,0,5,7,5,1,8,7,0,3,1,3,5,5,7,2,3,3,2,6,3,7,0,5,1,6,4,6,0,
%U A273093 0,7,3,6,0,3,1,0,6,7,9,3,2,6,2,5,3,6,5,9,3,0,3,5,9,1,0,6,6,0,4,9
%N A273093 Decimal expansion of the probability that three positive integers are pairwise not coprime.
%D A273093 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.5.1 Carefree Couples, p. 110.
%H A273093 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 15.
%H A273093 Randell Heyman, <a href="https://arxiv.org/abs/1309.5578">Pairwise non-coprimality of triples</a>, arXiv preprint, arXiv:1309.5578 [math.NT], 2013-2014.
%H A273093 Randell Heyman, <a href="http://unsworks.unsw.edu.au/fapi/datastream/unsworks:36429/SOURCE02?view=true">Topics in divisibility: pairwise coprimality, the GCD of shifted sets and polynomial irreducibility</a>, PhD thesis, University of New South Wales, 2015.
%H A273093 Pieter Moree, <a href="http://arxiv.org/abs/math/0510003">Counting carefree couples</a>, arXiv:math/0510003 [math.NT], 2005-2014.
%H A273093 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/CarefreeCouple.html">Carefree Couple</a>.
%F A273093 Equals 1 - 18/Pi^2 + 3P - Q, where P is A065464 and Q is A065473.
%e A273093 0.1742197830347247005585740721805346916511057518703135572332637051646...
%t A273093 $MaxExtraPrecision = 1000; digits = 100; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]];
%t A273093 P = (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n - 1]/(n - 1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]];
%t A273093 Q = NSum[-(2 + (-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 3digits, Method -> "AlternatingSigns"]//Exp;
%t A273093 F3 = 1 - 18/Pi^2 + 3P - Q;
%t A273093 RealDigits[F3, 10, digits][[1]]
%o A273093 (PARI) 1 - 3/zeta(2) + 3 * prodeulerrat(1 - (2*p-1)/p^3) - prodeulerrat(1 - (3*p-2)/p^3) \\ _Amiram Eldar_, Mar 03 2021
%Y A273093 Cf. A065464, A065473, A229099.
%K A273093 nonn,cons
%O A273093 0,2
%A A273093 _Jean-François Alcover_, May 15 2016