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A065300 Numbers k such that the sum of divisors of k is a squarefree number.

%I A065300 #39 Feb 24 2025 02:03:41
%S A065300 1,2,4,5,8,9,13,16,18,20,25,26,29,36,37,41,45,49,50,61,64,72,73,74,80,
%T A065300 100,101,104,109,113,116,117,121,122,128,137,144,146,148,157,169,173,
%U A065300 180,181,193,196,200,208,218,225,229,234,242,244,256,257,261,277,281
%N A065300 Numbers k such that the sum of divisors of k is a squarefree number.
%C A065300 Numbers k such that sigma(k) divides primorial(k), that is, A002110(k) mod A000203(k) = 0. - _Gary Detlefs_, May 02 2012
%H A065300 Amiram Eldar, <a href="/A065300/b065300.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%F A065300 Solutions to |mu(sigma(x))| = 1.
%e A065300 For k = 100, sigma(100) = 217 = 7*31.
%t A065300 Select[Range@ 300, SquareFreeQ@ DivisorSigma[1, #] &] (* or *)
%t A065300 Select[Range@ 300, Abs@ MoebiusMu@ DivisorSigma[1, #] == 1 &] (* _Michael De Vlieger_, Mar 18 2017 *)
%o A065300 (PARI) { n=0; for (m = 1, 10^9, if (abs(moebius(sigma(m)))==1, write("b065300.txt", n++, " ", m); if (n==1000, return)) ) } \\ _Harry J. Smith_, Oct 15 2009
%o A065300 (PARI) for(n=1, 300, if(issquarefree(sigma(n)), print1(n,", "))) \\ _Indranil Ghosh_, Mar 19 2017
%o A065300 (Python)
%o A065300 from sympy import mobius, divisor_sigma
%o A065300 from sympy.ntheory.factor_ import core
%o A065300 [n for n in range(1,301) if abs(mobius(divisor_sigma(n, 1))) == 1] #* or *#
%o A065300 [n for n in range(1,301) if core(divisor_sigma(n,1)) == divisor_sigma(n,1)] # _Indranil Ghosh_, Mar 19 2017
%Y A065300 Cf. A000203 (sigma), A002110, A005117, A008683 (mu), A065299.
%K A065300 nonn
%O A065300 1,2
%A A065300 _Labos Elemer_, Oct 29 2001