A065301 Both n 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

1,2

Links

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

Example

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

Mathematica

Select[Range[1000], AllTrue[{#, DivisorSigma[1, #]}, SquareFreeQ]&] (* Uses the function AllTrue from Mathematica version 10 *) (* Harvey P. Dale, Aug 09 2014 *)

Prog

(PARI) { n=0; for (m = 1, 10^9, if (abs(moebius(m))==1 && abs(moebius(sigma(m)))==1, write("b065301.txt", n++, " ", m); if (n==1000, return)) ) } \\ 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

Crossrefs

Cf. A000203, A008683, A065299, A065300.

Cf. A065303, A087248, A087249.

Sequence in context: A369714 A289529 A087250 * A326511 A289463 A289579

Adjacent sequences: A065298 A065299 A065300 * A065302 A065303 A065304

Keyword

nonn

Author

Labos Elemer, Oct 29 2001

Status

approved