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

6, 11, 18, 20, 27, 30, 37, 38, 51, 66, 67, 70, 83, 85, 102, 123, 128, 135, 146, 171, 198, 219, 227, 248, 258, 260, 264, 291, 326, 346, 363, 402, 443, 486, 515, 521, 531, 578, 627, 629, 678, 731, 732, 735, 786, 843, 902, 963, 1003, 1026, 1029, 1034, 1091, 1158

Offset: 1

Author

T. D. Noe, Oct 06 2004

Keywords

• nonn

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

Crossrefs

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).

Programs

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]