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
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
KEYWORD
nonn
AUTHOR
Martin Kochanski (mjk(AT)cardbox.com), May 10 2007
STATUS
approved