A065303 - OEIS

%I #28 May 16 2020 03:34:27

%S 12,24,27,28,32,40,44,48,52,54,56,60,63,68,75,76,81,84,88,90,92,96,98,

%T 99,108,112,120,124,125,126,132,135,136,140,147,150,152,153,156,160,

%U 162,164,168,171,172,175,176,184,188,189,192,198,204,207,212,216,220

%N Neither n nor sigma(n) is squarefree.

%H Harry J. Smith, <a href="/A065303/b065303.txt">Table of n, a(n) for n = 1..1000</a>

%e n = 147 = 3*7*7, sigma(147) = 2*2*3*19 = 228.

%t Select[Range@ 220, Nor[SquareFreeQ@ #, SquareFreeQ@ DivisorSigma[1, #]] &] (* _Michael De Vlieger_, Mar 18 2017 *)

%t Select[Range[250],NoneTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 22 2019 *)

%o (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

%o (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

%o 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

%o (Python)

%o from sympy import divisor_sigma

%o from sympy.ntheory.factor_ import core

%o def is_squarefree(n): return core(n) == n

%o 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

%Y Cf. A000203, A008683, A065299, A065200, A065201, A065302.

%K nonn

%O 1,1

%A _Labos Elemer_, Oct 29 2001