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

1,1

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?

Links

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

Example

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

Prog

(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)

Crossrefs

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

Sequence in context: A081948 A191273 A087361 * A090117 A329125 A020181

Adjacent sequences: A340639 A340640 A340641 * A340643 A340644 A340645

Keyword

nonn

Author

Hugo Pfoertner, Jan 14 2021

Status

approved