A087249 -id:A087249 - cp's OEIS frontend

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

Labos Elemer, Oct 29 2001

nonn

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)

			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.

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

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

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

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

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

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

A086368 Nonsquarefree numbers k such that sigma(k) is squarefree.

Original entry on oeis.org

4, 8, 9, 16, 18, 20, 25, 36, 45, 49, 50, 64, 72, 80, 100, 104, 116, 117, 121, 128, 144, 148, 169, 180, 196, 200, 208, 225, 234, 242, 244, 256, 261, 289, 292, 296, 320, 325, 333, 361, 369, 404, 436, 441, 450, 452, 464, 488, 512, 529, 548, 549, 576, 578, 584, 592

Offset: 1

Labos Elemer, Sep 05 2003

nonn

			k = 45 = 3*3*5 and sigma(45) = 78 = 2*3*13.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A013929 and A065300.

Cf. A000203 (sigma), A005117, A065303, A087249, A065301.

filter:= n -> not numtheory:-issqrfree(n) and numtheory:-issqrfree(numtheory:-sigma(n)):
select(filter, [$1..1000]); # Robert Israel, May 11 2018

q[k_] := ! SquareFreeQ[k] && SquareFreeQ[DivisorSigma[1, k]]; Select[Range[600], q] (* Amiram Eldar, Feb 24 2025 *)

isok(k) = !issquarefree(k) && issquarefree(sigma(k)); \\ Amiram Eldar, Feb 24 2025

cp's OEIS Frontend

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

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