A099226 Numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1.

27, 248, 2194, 32763

Offset

1,1

Comments

No other terms < 10^15. The intersection of A057897 and A099225. The representation question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x+y for y > x > 1 and b > a > 1.

Links

Table of n, a(n) for n=1..4.

Example

27 = 25^2+2 = 32^5-5, 248 = 7^3+3 = 2^8-8, 2194 = 3^7+7 = 13^3-3 and 32763 = 181^2+2 = 8^5-5.

Mathematica

nLim=40000; lst1={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst1, n]; k++ ], {m, 2, Sqrt[nLim]}]; lst2={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst2, n]; k++ ], {m, 2, Sqrt[nLim]}]; Intersection[lst1, lst2]

Crossrefs

Cf. A074981 (n such that there is no solution to Pillai's equation).

Sequence in context: A074117 A030509 A042414 * A016107 A101379 A132652

Adjacent sequences: A099223 A099224 A099225 * A099227 A099228 A099229

Keyword

nonn,more

Author

T. D. Noe, Oct 06 2004

Status

approved