

A219246


The decimal expansion of the maximum M(5) of the ratio (sum{k=1,...,5} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.


5



1, 4, 8, 6, 3, 5, 3, 2, 2, 8, 9, 6, 3, 0, 5, 0, 6, 4, 0, 5, 2, 0, 4, 8, 7, 1, 6, 4, 6, 1, 9, 8, 5, 1, 5, 6, 6, 4, 3, 5, 4, 6, 9, 5, 6, 4, 1, 0, 0, 9, 3, 7, 9, 4, 5, 3, 2, 5, 3, 3, 5, 5, 8, 8, 2, 3, 9, 8, 9, 3, 8, 1, 0, 1, 4, 8, 1, 5, 9, 8, 7, 5, 5, 6, 6, 2, 4, 1, 9, 0, 0, 7, 4, 6, 1, 1, 3, 2, 2, 4, 4, 7
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OFFSET

1,2


COMMENTS

The maximum M(n) of the ratio (sum{k=1,...,n} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(n)) taken over x(1), ..., x(n) > 0 is discussed in A219245  see also the paper of Witula et al. for the proofs.
The decimal expansions of M(4) and M(6) in A219245 and A219336 respectively are presented.


REFERENCES

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), 9396.


LINKS

Table of n, a(n) for n=1..102.
Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
YuDong Wu, ZhiHua Zhang and ZhiGang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008


EXAMPLE

1.486353228963....


MATHEMATICA

RealDigits[c5/.FindRoot[{1+x2/2+x3/3+x4/4+x5/5==c5, x2/2+x3/3+x4/4+x5/5==c5*x2^2, x3/3+x4/4+x5/5==c5*x3^3/x2^2, x4/4+x5/5==c5*x4^4/x3^3, x5/5==c5*x5^5/x4^4}, {{c5, 3/2}, {x2, 1/2}, {x3, 1/2}, {x4, 1/2}, {x5, 1/2}}, WorkingPrecision>120], 10, 105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)


CROSSREFS

Cf. A219245, A219336, A249403.
Sequence in context: A175475 A193082 A201335 * A296488 A199294 A155741
Adjacent sequences: A219243 A219244 A219245 * A219247 A219248 A219249


KEYWORD

nonn,cons


AUTHOR

Roman Witula, Nov 16 2012


STATUS

approved



