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

1,1

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

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

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

Sequence in context: A095780 A119588 A167994 * A239673 A372548 A208158

Adjacent sequences: A065300 A065301 A065302 * A065304 A065305 A065306

Keyword

nonn

Author

Labos Elemer, Oct 29 2001

Status

approved