A102834 Numbers whose factors are primes raised to powers >= 2 and are not perfect squares.

8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 864, 968, 972, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 1944, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2744, 2888, 3087, 3125, 3200, 3267, 3375

Offset: 1

Author

Cino Hilliard, Feb 27 2005

Keywords

• easy
• nonn

Comments

Powerful numbers (A001694) that are not perfect squares. - T. D. Noe, May 03 2006

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A001694.

Programs

Mathematica

Powerful[n_Integer] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Select[Range[10000], Powerful[ # ] && !IntegerQ[Sqrt[ # ]]&] (* T. D. Noe, May 03 2006 *)

PARI

omnipnotsq(n,m)= local(a,x,j,fl=0); for(x=1,n, a=factor(x); for(j=1,omega(x), if(a[j,2]>= m,fl=1,fl=0;break); ); if(fl&issquare(x)==0,print1(x",")) )

PARI

is(n)=ispowerful(n) && !issquare(n) \\ Charles R Greathouse IV, Oct 19 2015

Python

from math import isqrt
from sympy import integer_nthroot, mobius
def A102834(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j = isqrt(x)
        c, l = n+x+j, 0
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Formula

Sum_{n>=1} 1/a(n)^s = zeta(2*s)*(zeta(3*s)/zeta(6*s) - 1), s > 1/2. - Amiram Eldar, Apr 06 2023