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

1,2

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)

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

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

Sequence in context: A369714 A289529 A087250 * A326511 A396392 A289463

Adjacent sequences: A065298 A065299 A065300 * A065302 A065303 A065304

Keyword

nonn

Author

Labos Elemer, Oct 29 2001

Status

approved