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

1,1

Comments

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

Links

Table of n, a(n) for n=1..44.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A153326 A368959 A338325 * A182046 A171468 A114377

Adjacent sequences: A168360 A168361 A168362 * A168364 A168365 A168366

Keyword

nonn

Author

Franklin T. Adams-Watters, Nov 23 2009

Status

approved