OFFSET

1,1

COMMENTS

For n=11, there are two representations: 2^3+3 and 3^2+2. All other numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = y-x for y > x > 1 and b > a > 1, which appears to have only one solution.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

N:= 2000: # for terms <= N

S:= {}:

for k from 2 to floor(log[2](N)) do

S:= S union {seq(m^k+k, m=2..floor((N-k)^(1/k)))}

od:

sort(convert(S, list)); # Robert Israel, Apr 28 2019

MATHEMATICA

nLim=2000; lst={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst]

CROSSREFS

KEYWORD

nonn

AUTHOR

T. D. Noe, Oct 06 2004

STATUS

approved