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 A085849 #38 Feb 16 2025 08:32:50
%S A085849 3,5,3,2,3,6,3,7,1,8,5,4,9,9,5,9,8,4,5,4,3,5,1,6,5,5,0,4,3,2,6,8,2,0,
%T A085849 1,1,2,8,0,1,6,4,7,7,8,5,6,6,6,9,0,4,4,6,4,1,6,0,8,5,9,4,2,8,1,4,2,3,
%U A085849 8,3,2,5,0,0,2,6,6,9,0,0,3,4,8,3,6,7,2,0,7,8,3,3,4,3,3,5,4,9,8,9,6,7
%N A085849 Decimal expansion of the probability that two m X m and n X n matrices (m,n large) have relatively prime determinants.
%C A085849 The Hafner-Sarnak-McCurley constant. [Named after the American mathematician James Lee Hafner (1954-2015), the South-African and American mathematician Peter Clive Sarnak (b. 1953) and the American mathematician and computer scientist Kevin Snow McCurley. - _Amiram Eldar_, Jun 15 2021]
%C A085849 Comment on numerics from _R. J. Mathar_, Apr 20 2011: (Start)
%C A085849 The definition s = Product_{p} (1-[1- Product_{n>=1} (1-1/p^n)]^2) may be binomially expanded to s = Product_{p} Sum_{n>=1} (2*A010815(n)-A002107(n))/p^n. The auxiliary sequence 2*A010815(n)-A002107(n) is 1, 0, -1, -2, -1, 0, 2, 2, 2, 2, -1, 0,... for n>=0.
%C A085849 The inverse Euler transformation of the auxiliary sequence generates Sum_{n} (2*A010815(n)-A002107(n)) /p^n = Product_{n} (1-1/p^n)^gamma(n) with gamma(n) = 0, -1, -2 ,-1, -2, 0, -2, -1, 0, -2, 0, -1,... for n>=1. This yields s = Product_{n>=1} zeta(n)^gamma(n) where zeta(n) are the values of the Riemann zeta function.
%C A085849 (End)
%D A085849 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.5, "Hafner-Sarnak-McCurley Constant", pp. 110-112.
%D A085849 Ilan Vardi, Computational Recreations in Mathematica, Redwood City, CA: Addison-Wesley, 1991, p. 174.
%H A085849 Philippe Flajolet and Ilan Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta function expansions of some classical constants</a>.
%H A085849 J. L. Hafner, P. Sarnak and K. McCurley, <a href="http://www.mccurley.org/papers/relprime.pdf">Relatively prime values of polynomials</a>, in: M. Knopp and M. Sheigorn, Editors, A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary Mathematics, Vol. 143, AMS, 1993.
%H A085849 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hafner-Sarnak-McCurleyConstant.html">Hafner-Sarnak-McCurley Constant</a>.
%H A085849 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hafner-Sarnak-McCurley_constant">Hafner-Sarnak-McCurley constant</a>.
%F A085849 Equals Product_{p prime} (1-(1-Product_{n>=1} (1-1/p^n))^2). - _Benoit Cloitre_, Aug 05 2003
%e A085849 0.3532363718549959845435165504326820112801647785666904464160859428...
%t A085849 digits = 102; CC = CoefficientList[Log[1 - (1 - QPochhammer[1/p])^2] + O[p, Infinity]^(4 digits), 1/p][[3 ;; -1]]; Hafner = CC.Table[PrimeZetaP[n + 1], {n, 1, Length[CC]}] // Exp // N[#, digits+10]&; RealDigits[Hafner, 10, digits][[1]] (* _Jean-François Alcover_, Apr 25 2016 *)
%Y A085849 Cf. A002107, A010815.
%K A085849 nonn,cons
%O A085849 0,1
%A A085849 _Eric W. Weisstein_, Jul 05 2003
%E A085849 More terms from _Benoit Cloitre_, Aug 05 2003
%E A085849 Edited by _N. J. A. Sloane_, Feb 11 2009 at the suggestion of _R. J. Mathar_
%E A085849 Twenty additional digits from _R. J. Mathar_, Feb 13 2009
%E A085849 Extended to 100 digits by _Jean-François Alcover_, Apr 25 2016