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

%I A065301 #35 Mar 08 2025 06:00:48
%S A065301 1,2,5,13,26,29,37,41,61,73,74,101,109,113,122,137,146,157,173,181,
%T A065301 193,218,229,257,277,281,313,314,317,353,362,373,386,389,397,401,409,
%U A065301 421,433,457,458,461,509,541,554,569,601,613,617,626,641,653,661,673,677
%N A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.
%C A065301 From _Amiram Eldar_, Mar 08 2025: (Start)
%C A065301 Number k such that A280710(k) * A280710(A000203(k)) = 1, or equivalently, A280710(k) * A280710(A048250(k)) = 1.
%C A065301 Squarefree numbers k whose prime factors are terms of A049097, and the elements of the set {p+1 , p|k} are pairwise coprime. (End)
%H A065301 Amiram Eldar, <a href="/A065301/b065301.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%e A065301 For k = 13, sigma(13) = 14 = 2*7 is squarefree.
%e A065301 For k = 26, sigma(26) = 1 + 2 + 13 + 26 = 42 = 2*3*7 is squarefree.
%e A065301 For k = 277 (prime), sigma(277) = 278 = 2*139 is squarefree.
%t A065301 Select[Range[1000],AllTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* _Harvey P. Dale_, Aug 09 2014 *)
%o A065301 (PARI) is(m) = abs(moebius(m))==1 && abs(moebius(sigma(m)))==1 \\ _Harry J. Smith_, Oct 15 2009
%o A065301 (Python)
%o A065301 from sympy import divisor_sigma
%o A065301 from sympy.ntheory.factor_ import core
%o A065301 def issquarefree(n): return core(n)==n
%o A065301 print([n for n in range(1, 1001) if issquarefree(n) and issquarefree(divisor_sigma(n,1))]) # _Indranil Ghosh_, Mar 19 2017
%Y A065301 Cf. A000203, A008683, A048250, A065299, A065300.
%Y A065301 Cf. A049097, A065303, A087248, A087249, A280710.
%K A065301 nonn
%O A065301 1,2
%A A065301 _Labos Elemer_, Oct 29 2001