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A065303
Neither n nor sigma(n) is squarefree.
6
12, 24, 27, 28, 32, 40, 44, 48, 52, 54, 56, 60, 63, 68, 75, 76, 81, 84, 88, 90, 92, 96, 98, 99, 108, 112, 120, 124, 125, 126, 132, 135, 136, 140, 147, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 198, 204, 207, 212, 216, 220
OFFSET
1,1
LINKS
EXAMPLE
n = 147 = 3*7*7, sigma(147) = 2*2*3*19 = 228.
MATHEMATICA
Select[Range@ 220, Nor[SquareFreeQ@ #, SquareFreeQ@ DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 18 2017 *)
Select[Range[250], NoneTrue[{#, DivisorSigma[1, #]}, SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2019 *)
PROG
(PARI) n=0; for (m = 1, 10^9, if (!moebius(m) && !moebius(sigma(m)), write("b065303.txt", n++, " ", m); if (n==1000, return)) ) \\ Harry J. Smith, Oct 16 2009
(PARI) sigmaSquarefree(f)=my(v=vector(#f~, i, (f[i, 1]^(f[i, 2]+1)-1) / (f[i, 1]-1))); for(i=2, #v, for(j=1, i-1, if(gcd(v[i], v[j])>1, return(0)))); for(i=1, #v, if(!issquarefree(v[i]), return(0))); 1
list(lim)=my(v=List()); forfactored(k=12, lim\1, if(!issquarefree(k) && !sigmaSquarefree(k[2]), listput(v, k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018
(Python)
from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
def is_squarefree(n): return core(n) == n
print([i for i in range(1, 251) if not is_squarefree(i) and not is_squarefree(divisor_sigma(i, 1))]) # Indranil Ghosh, Mar 18 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Oct 29 2001
STATUS
approved