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A340643
Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 2.
2
2, 3, 5, 15, 26, 46, 82, 89, 90, 129, 323, 362, 401, 420, 610, 624, 840, 2024, 2703, 2808, 6888, 12099, 15963, 19320, 24650, 29930, 33490, 36482, 39203, 45795, 47523, 52440, 66050, 69168, 83408, 94248, 94863, 103683, 114284, 164399, 185364, 206442, 222785, 227530, 229180
OFFSET
1,1
COMMENTS
Within the range of the data, a(n)^2 = A340642(n), i.e., no 3 immediately consecutive perfect powers x^p1, y^p2, z^p3 with min (p1, p2, p3) > 2 are seen. Is there a counterexample?
LINKS
David A. Corneth, Table of n, a(n) for n = 1..611 (first 181 terms from Hugo Pfoertner)
PROG
(PARI) a340643(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(issquare(n1)&p2>2&p0>2, print1(sqrtint(n1), ", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340643(10^8)
(PARI) upto(n) = {n *= n; my(v = List(), res = List([2])); for(i = 2, sqrtnint(n, 3), for(e = 3, logint(n, i), listput(v, i^e) ); ); listsort(v, 1); for(i = 1, #v - 1, if(sqrtint(v[i]) + 1 == sqrtint(v[i+1]) - issquare(v[i+1]), listput(res, sqrtint(v[i+1]-issquare(v[i+1]))); ) ); res }
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jan 14 2021
EXTENSIONS
More terms from David A. Corneth, Jan 14 2021
STATUS
approved