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A102834
Numbers whose factors are primes raised to powers >= 2 and are not perfect squares.
5
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
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
Sequence in context: A070265 A354179 A262675 * A376171 A370786 A116002
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Feb 27 2005
STATUS
approved