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Search: id:A111334
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| A111334 |
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a(k) is the smallest integer n with the property that the difference between the arithmetic and the geometric mean of the first n numbers is bigger than 10^k. |
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1
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| 11, 81, 765, 7581, 75703, 756903, 7568866, 75688472, 756884504, 7568844796, 75688447681, 756884476508, 7568844764750, 75688447647137, 756884476470980, 7568844764709381, 75688447647093366, 756884476470933182
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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By using the approximation formula n! = (n/e)^n one can show that a(k) will be approximately 7.56*10^k.
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FORMULA
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the smallest n for which ((n+1)/2 - (n!)^(1/n)) > 10^k
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EXAMPLE
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(80+1)/2 - (80!)^(1/80) = 9.9026... < 10^1 < 10.032.. = (81+1)/2 - (81!)^(1/81)
So 81 is the smallest n where the required difference exceeds 10, thus a(1) = 81.
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PROGRAM
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(PARI) f(n)=return(log(sqrt(2*Pi))+(n+0.5)*log(n)-n+1/(12*n)) for(k=0, 24, n=0; forstep(i=4*k+8, 0, -1, m=n+2^i; \ if(f(m)>m*log((m+1)/2-10^k), n=m)); print1(n+1, ", ")) - Robert Gerbicz (robert.gerbicz(AT)gmail.com), Aug 24 2006
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CROSSREFS
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Sequence in context: A119364 A055429 A003730 this_sequence A085879 A029887 A026871
Adjacent sequences: A111331 A111332 A111333 this_sequence A111335 A111336 A111337
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KEYWORD
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nonn
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AUTHOR
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Stefan Steinerberger (hansibal(AT)hotmail.com), Nov 05 2005
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EXTENSIONS
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More terms from Robert Gerbicz (robert.gerbicz(AT)gmail.com), Aug 24 2006
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