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A128427 Last point where sum of n consecutive n-th powers does not exceed the next n-th power. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=2..71.

M. J. Kochanski, Cyprian's Last Theorem (a work in progress).

FORMULA

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.

EXAMPLE

a(2)=5 because 3^2+4^2=5^2; a(3)=6 because 3^3+4^3+5^3=6^3.

CROSSREFS

Sequence in context: A288857 A201472 A005049 * A292917 A120182 A037361

Adjacent sequences:  A128424 A128425 A128426 * A128428 A128429 A128430

KEYWORD

nonn

AUTHOR

Martin Kochanski (mjk(AT)cardbox.com), May 10 2007

STATUS

approved

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Last modified December 6 17:05 EST 2021. Contains 349567 sequences. (Running on oeis4.)