A340643 Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 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

Author

Hugo Pfoertner, Jan 14 2021

Keywords

• nonn

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)

Crossrefs

Cf. A000290, A001597, A025479, A076467, A097054, A111245, A153158, A340642, A340700, A340701.

Programs

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 }

Extensions

More terms from David A. Corneth, Jan 14 2021