cp's OEIS Frontend

A099225 Numbers of the form m^k+k, with m and k > 1.

%I A099225 #11 Feb 01 2021 18:18:52
%S A099225 6,11,18,20,27,30,37,38,51,66,67,70,83,85,102,123,128,135,146,171,198,
%T A099225 219,227,248,258,260,264,291,326,346,363,402,443,486,515,521,531,578,
%U A099225 627,629,678,731,732,735,786,843,902,963,1003,1026,1029,1034,1091,1158
%N A099225 Numbers of the form m^k+k, with m and k > 1.
%C A099225 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.
%H A099225 Robert Israel, <a href="/A099225/b099225.txt">Table of n, a(n) for n = 1..10000</a>
%p A099225 N:= 2000: # for terms <= N
%p A099225 S:= {}:
%p A099225 for k from 2 to floor(log[2](N)) do
%p A099225   S:= S union {seq(m^k+k, m=2..floor((N-k)^(1/k)))}
%p A099225 od:
%p A099225 sort(convert(S,list)); # _Robert Israel_, Apr 28 2019
%t A099225 nLim=2000; lst={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst]
%Y A099225 Cf. A057897 (numbers of the form m^k-k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).
%K A099225 nonn
%O A099225 1,1
%A A099225 _T. D. Noe_, Oct 06 2004