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A219336 The decimal expansion of the maximum M(6) of the ratio (sum{k=1,...,6} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 0. 5
1, 5, 3, 7, 9, 3, 7, 5, 5, 6, 5, 2, 0, 0, 3, 4, 9, 3, 1, 3, 6, 8, 1, 5, 8, 7, 1, 6, 0, 2, 6, 3, 2, 6, 8, 1, 5, 6, 0, 8, 6, 4, 5, 0, 8, 9, 8, 6, 3, 2, 1, 9, 6, 3, 3, 3, 2, 4, 6, 4, 3, 1, 1, 6, 3, 0, 0, 9, 2, 7, 6, 4, 1, 4, 2, 6, 1, 2, 9, 3, 4, 2, 5, 2, 3, 7, 7, 9, 3, 8, 0, 1, 3, 1, 4, 4, 2, 2, 9, 9, 5, 1, 9 (list; constant; graph; refs; listen; history; text; internal format)
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 the M(4) and M(5) in A219245 and A219246 respectively are given.

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), 93-96.

LINKS

Table of n, a(n) for n=1..103.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]

Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008

EXAMPLE

1.537937556520034931368158716...

MATHEMATICA

RealDigits[c6/.FindRoot[{1 + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6, x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6*x2^2, x3/3 + x4/4 + x5/5 + x6/6 == c6*x3^3/x2^2, x4/4 + x5/5 + x6/6 == c6*x4^4/x3^3, x5/5 + x6/6 == c6*x5^5/x4^4, x6/6 == c6*x6^6/x5^5}, {{c6, 3/2}, {x2, 1/2}, {x3, 1/2}, {x4, 1/2}, {x5, 1/2}, {x6, 1/2}}, WorkingPrecision->120], 10, 105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)

CROSSREFS

Cf. A219245, A219246, A249403.

Sequence in context: A186905 A109694 A259068 * A280235 A135765 A222598

Adjacent sequences:  A219333 A219334 A219335 * A219337 A219338 A219339

KEYWORD

nonn,cons

AUTHOR

Roman Witula, Nov 18 2012

STATUS

approved

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Last modified August 18 04:50 EDT 2019. Contains 326072 sequences. (Running on oeis4.)