A065303 Neither n nor sigma(n) is squarefree.
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
Keywords
Examples
n = 147 = 3*7*7, sigma(147) = 2*2*3*19 = 228.
Links
- Harry J. Smith, Table of n, a(n) for n = 1..1000
Programs
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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 *)
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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
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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