A340643 - OEIS

%I #27 Nov 04 2023 12:32:06

%S 2,3,5,15,26,46,82,89,90,129,323,362,401,420,610,624,840,2024,2703,

%T 2808,6888,12099,15963,19320,24650,29930,33490,36482,39203,45795,

%U 47523,52440,66050,69168,83408,94248,94863,103683,114284,164399,185364,206442,222785,227530,229180

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

%C 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?

%H David A. Corneth, <a href="/A340643/b340643.txt">Table of n, a(n) for n = 1..611</a> (first 181 terms from Hugo Pfoertner)

%o (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))};

%o a340643(10^8)

%o (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 }

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

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Jan 14 2021

%E More terms from _David A. Corneth_, Jan 14 2021