OFFSET
0,2
COMMENTS
The two trivial cases of equality are n=0, m=1 and n=1, m=2, i.e. 1^0=2^0 and 1^1+2^1=3^1. The references state that there are no other equalities for m<10^2000000.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, D10.
LINKS
Eric Weisstein's World of Mathematics, Power.
FORMULA
Conjecture: a(n) = 1 + round(n/log(2)). Formula verified for n=1..700. - Herbert Kociemba, Apr 08 2020
EXAMPLE
a(3)=5 since 1^3+2^3+3^3+4^3<5^3 but 1^3+2^3+3^3+4^3+5^3>=6^3, i.e. since 100<125 but 225>=216.
MAPLE
A072633 := proc(n)
local msum, m ;
msum := 1;
m := 1 ;
while msum < (m+1)^n do
m := m+1 ;
msum := msum+m^n ;
end do:
return m ;
end proc:
seq(A072633(n), n=0..30) ; # R. J. Mathar, Feb 27 2018
MATHEMATICA
(* Assuming sequence is increasing : *) a[0] = 1; a[n_] := a[n] = (m = a[n-1]; While[ True, m++; If[ Sum[ k^n, {k, 1, m}] >= (m+1)^n, Break[]]]; m); Table[ a[n], {n, 0, 69}] (* Jean-François Alcover, Oct 03 2011 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, Jun 28 2002
STATUS
approved