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A128427
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Last point where sum of n consecutive n-th powers does not exceed the next n-th power.
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0
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5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 103
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OFFSET
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2,1
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COMMENTS
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Denoting by x(n) the largest real root of (x-n)^n+...+(x-1)^n=x^N (so that by definition a(n)=floor(x(n))), it is conjectured ("Cyprian's Last Theorem") but not fully proved that x(n) is never an integer for n>3. A (very rapidly very good) asymptotic approximation to x(n) is 1.5 + n / ln2 + O(1/n), but this needs proof. This yields floor(1.5 + n / ln2) as an approximation to a(n).
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LINKS
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Table of n, a(n) for n=2..71.
M. J. Kochanski, Cyprian's Last Theorem (a work in progress).
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FORMULA
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a(n) is the largest integer for which the sum of the n consecutive n-th powers from (a(n)-n)^n to (a(n)-1)^n inclusive does not exceed a(n)^n.
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EXAMPLE
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a(2)=5 because 3^2+4^2=5^2; a(3)=6 because 3^3+4^3+5^3=6^3.
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CROSSREFS
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Sequence in context: A088721 A201472 A005049 * A120182 A037361 A202014
Adjacent sequences: A128424 A128425 A128426 * A128428 A128429 A128430
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KEYWORD
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nonn
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AUTHOR
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Martin Kochanski (mjk(AT)cardbox.com), May 10 2007
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STATUS
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approved
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