A065300 Numbers k such that the sum of divisors of k is a squarefree number.

1, 2, 4, 5, 8, 9, 13, 16, 18, 20, 25, 26, 29, 36, 37, 41, 45, 49, 50, 61, 64, 72, 73, 74, 80, 100, 101, 104, 109, 113, 116, 117, 121, 122, 128, 137, 144, 146, 148, 157, 169, 173, 180, 181, 193, 196, 200, 208, 218, 225, 229, 234, 242, 244, 256, 257, 261, 277, 281

Offset: 1

Author

Labos Elemer, Oct 29 2001

Keywords

• nonn

Comments

Numbers k such that sigma(k) divides primorial(k), that is, A002110(k) mod A000203(k) = 0. - Gary Detlefs, May 02 2012

Examples

			For k = 100, sigma(100) = 217 = 7*31.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A000203 (sigma), A002110, A005117, A008683 (mu), A065299.

Programs

Mathematica

Select[Range@ 300, SquareFreeQ@ DivisorSigma[1, #] &] (* or *)
Select[Range@ 300, Abs@ MoebiusMu@ DivisorSigma[1, #] == 1 &] (* Michael De Vlieger, Mar 18 2017 *)

PARI

{ n=0; for (m = 1, 10^9, if (abs(moebius(sigma(m)))==1, write("b065300.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 15 2009

PARI

for(n=1, 300, if(issquarefree(sigma(n)), print1(n,", "))) \\ Indranil Ghosh, Mar 19 2017

Python

from sympy import mobius, divisor_sigma
from sympy.ntheory.factor_ import core
[n for n in range(1,301) if abs(mobius(divisor_sigma(n, 1))) == 1] #* or *#
[n for n in range(1,301) if core(divisor_sigma(n,1)) == divisor_sigma(n,1)] # Indranil Ghosh, Mar 19 2017

Formula

Solutions to |mu(sigma(x))| = 1.