A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

4, 9, 25, 225, 676, 2116, 6724, 7921, 8100, 16641, 104329, 131044, 160801, 176400, 372100, 389376, 705600, 4096576, 7306209, 7884864, 47444544, 146385801, 254817369, 373262400, 607622500, 895804900, 1121580100, 1330936324, 1536875209, 2097182025, 2258435529, 2749953600

Offset: 1

Author

Hugo Pfoertner, Jan 14 2021

Keywords

• nonn

Comments

Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?

Examples

			The first terms, assuming 1 being at least a cube:
.
  n   p1  x^p1  p2  a(n)  p3  z^p3
                   =y^p2
  1  >2     1   2     4   3     8
  2   3     8   2     9   4    16
  3   4    16   2    25   3    27
  4   3   216   2   225   5   243
  5   4   625   2   676   6   729
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..181 (terms < 10^18)

Crossrefs

Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.

Programs

PARI

a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340642(50000000)