A168363 Squares and cubes of primes.

4, 8, 9, 25, 27, 49, 121, 125, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 17161, 18769, 19321, 22201

Offset: 1

Author

Franklin T. Adams-Watters, Nov 23 2009

Keywords

• nonn

Comments

Primitive elements for powerful numbers; every powerful is product of these numbers. The representation is not necessarily unique.

Crossrefs

Cf. A001694, A053810, A001248, A030078, A087797, A178254.

Programs

Mathematica

m=30000;Union[Prime[Range[PrimePi[m^(1/2)]]]^2,Prime[Range[PrimePi[m^(1/3)]]]^3] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
With[{nn=50},Take[Union[Flatten[Table[{n^2,n^3},{n,Prime[Range[ nn]]}]]],nn]] (* Harvey P. Dale, Feb 26 2015 *)

PARI

for(n=1,40000,fm=factor(n);if(matsize(fm)[1]==1&(fm[1,2]==2||fm[1,2]==3),print1(n",")))

PARI

is(n)=my(k=isprimepower(n)); k && k<4 \\ Charles R Greathouse IV, May 24 2013

Python

from math import isqrt
from sympy import primepi, integer_nthroot
def A168363(n):
    def f(x): return n+x-primepi(isqrt(x))-primepi(integer_nthroot(x,3)[0])
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

Formula

A178254(a(n)) = 2. - Reinhard Zumkeller, May 24 2010

Sum_{n>=1} 1/a(n) = P(2) + P(3) = 0.6270100593..., where P is the prime zeta function. - Amiram Eldar, Dec 21 2020