A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.

1, 2, 5, 13, 26, 29, 37, 41, 61, 73, 74, 101, 109, 113, 122, 137, 146, 157, 173, 181, 193, 218, 229, 257, 277, 281, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 541, 554, 569, 601, 613, 617, 626, 641, 653, 661, 673, 677

Offset: 1

Author

Labos Elemer, Oct 29 2001

Keywords

• nonn

Comments

From Amiram Eldar, Mar 08 2025: (Start)

Number k such that A280710(k) * A280710(A000203(k)) = 1, or equivalently, A280710(k) * A280710(A048250(k)) = 1.

Squarefree numbers k whose prime factors are terms of A049097, and the elements of the set {p+1 , p|k} are pairwise coprime. (End)

Examples

			For k = 13, sigma(13) = 14 = 2*7 is squarefree.
For k = 26, sigma(26) = 1 + 2 + 13 + 26 = 42 = 2*3*7 is squarefree.
For k = 277 (prime), sigma(277) = 278 = 2*139 is squarefree.
		

Links

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

Crossrefs

Cf. A000203, A008683, A048250, A065299, A065300.

Cf. A049097, A065303, A087248, A087249, A280710.

Programs

Mathematica

Select[Range[1000],AllTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* Harvey P. Dale, Aug 09 2014 *)

PARI

is(m) = abs(moebius(m))==1 && abs(moebius(sigma(m)))==1 \\ Harry J. Smith, Oct 15 2009

Python

from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n)==n
print([n for n in range(1, 1001) if issquarefree(n) and issquarefree(divisor_sigma(n,1))]) # Indranil Ghosh, Mar 19 2017