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 A249403 #20 Jul 01 2023 14:34:25
%S A249403 1,5,8,0,0,3,7,2,1,0,6,3,2,0,5,2,3,5,2,0,8,4,0,6,3,4,9,8,1,8,3,2,6,4,
%T A249403 4,9,2,1,1,2,8,1,5,8,0,5,9,1,6,5,9,6,1,9,7,0,1,7,4,2,3,6,9,2,0,6,0,1,
%U A249403 5,3,7,3,7,1,0,5,3,7,7,1,1,3,5,9,2,3,5,6,4,8,0,9,0,2,1,7,0,1,4,4,8,7,0,9,0
%N A249403 Decimal expansion of the maximum M(7) of the ratio (Sum_{k=1..7} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(7)) taken over x(1), ..., x(7) > 0.
%C A249403 M(2) = (1+sqrt(2))/2, M(3) = 4/3.
%C A249403 M(n) = exp(1) - 2*Pi^2*exp(1)/(log(n))^2 + O(1/(log(n))^3), [de Bruijn, 1963].
%D A249403 N. G. de Bruijn, Carleman's inequality for finite series, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag, Math., 25:505-514, 1963.
%D A249403 R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.
%H A249403 Steven R. Finch, <a href="/A219245/a219245.pdf">Carleman's inequality</a>, 2013. [Cached copy, with permission of the author]
%H A249403 Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, <a href="http://www.emis.de/journals/AMAPN/vol24_2/7.html">The Best Constant for Carleman's Inequality of Finite Type</a>, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008.
%e A249403 1.5800372106320523520840634981832644921128158059165961970174236920601537371...
%t A249403 RealDigits[c7/.FindRoot[{1 + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7, x2/2 + x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7*x2^2, x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7*x3^3/x2^2, x4/4 + x5/5 + x6/6 + x7/7 == c7*x4^4/x3^3, x5/5 + x6/6 + x7/7 == c7*x5^5/x4^4, x6/6 + x7/7 == c7*x6^6/x5^5, x7/7 == c7*x7^7/x6^6}, {{c7, 3/2}, {x2, 1/2}, {x3, 1/2}, {x4, 1/2}, {x5, 1/2}, {x6, 1/2}, {x7, 1/2}}, WorkingPrecision->120], 10, 105][[1]]
%Y A249403 Cf. A174968 = M(2), A219245 = M(4), A219246 = M(5), A219336 = M(6).
%K A249403 nonn,cons
%O A249403 1,2
%A A249403 _Vaclav Kotesovec_, Oct 27 2014