A097055 Numbers n such that the interval n^2 < x < (n+1)^2 contains at least one nonsquare perfect power A097054.

2, 5, 11, 14, 15, 18, 22, 31, 36, 41, 45, 46, 52, 55, 58, 70, 76, 82, 88, 89, 90, 96, 103, 110, 117, 129, 132, 140, 148, 156, 164, 172, 181, 189, 198, 207, 225, 234, 243, 252, 262, 272, 279, 281, 291, 301, 311, 316, 322, 332, 353, 362, 364, 374, 385, 396, 401

Offset: 1

Author

Hugo Pfoertner, Jul 21 2004

Keywords

• nonn

Examples

			a(1)=2 because 2^2<2^3<3^2, a(2)=5: 5^2<3^3<2^5<6^2, a(3)=11: 11^2<5^3<2^7<12^2.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A097054, A097056.

Programs

Mathematica

nn=1000^2; pp=Select[Union[Flatten[Table[n^i, {i,Prime[Range[2,PrimePi[Log[2,nn]]]]}, {n,2,nn^(1/i)}]]], !IntegerQ[Sqrt[#]]&]; Union[Floor[Sqrt[pp]]] (* T. D. Noe, Apr 19 2011 *)

PARI

is(n)=my(n2=n^2,t=n2+2*n); for(e=3,logint(t,2), if(sqrtnint(t,e)^e>n2, return(1))); 0 \\ Charles R Greathouse IV, Aug 28 2016

Python

from itertools import count, islice
from sympy import mobius, integer_nthroot
def A097055_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n: sum(mobius(k)*(1-integer_nthroot((n+1)**2,k)[0]) for k in range((n**2).bit_length(),((n+1)**2).bit_length()))+sum(mobius(k)*(integer_nthroot(n**2,k)[0]-integer_nthroot((n+1)**2,k)[0]) for k in range(3,(n**2).bit_length())), count(max(startvalue,2)))
A097055_list = list(islice(A097055_gen(),30)) # Chai Wah Wu, Aug 14 2024

Formula

a(n) ~ n^(3/2). - Charles R Greathouse IV, Aug 28 2016