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

1,1

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

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000290, A001694.

Sequence in context: A070265 A354179 A262675 * A376171 A370786 A116002

Adjacent sequences: A102831 A102832 A102833 * A102835 A102836 A102837

Keyword

easy,nonn,changed

Author

Cino Hilliard, Feb 27 2005

Status

approved