A065300 Numbers n such that sum of divisors 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

1,2

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000203, A008683, A065299.

Sequence in context: A101881 A143989 A064573 * A080403 A280050 A360772

Adjacent sequences: A065297 A065298 A065299 * A065301 A065302 A065303

Keyword

nonn

Author

Labos Elemer, Oct 29 2001

Status

approved