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A340642
Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.
5
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)
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jan 14 2021
STATUS
approved