A065300 - OEIS

%I #39 Feb 24 2025 02:03:41

%S 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 100,101,104,109,113,116,117,121,122,128,137,144,146,148,157,169,173,

%U 180,181,193,196,200,208,218,225,229,234,242,244,256,257,261,277,281

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

%C Numbers k such that sigma(k) divides primorial(k), that is, A002110(k) mod A000203(k) = 0. - _Gary Detlefs_, May 02 2012

%H 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 Solutions to |mu(sigma(x))| = 1.

%e For k = 100, sigma(100) = 217 = 7*31.

%t Select[Range@ 300, SquareFreeQ@ DivisorSigma[1, #] &] (* or *)

%t Select[Range@ 300, Abs@ MoebiusMu@ DivisorSigma[1, #] == 1 &] (* _Michael De Vlieger_, Mar 18 2017 *)

%o (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 (PARI) for(n=1, 300, if(issquarefree(sigma(n)), print1(n,", "))) \\ _Indranil Ghosh_, Mar 19 2017

%o (Python)

%o from sympy import mobius, divisor_sigma

%o from sympy.ntheory.factor_ import core

%o [n for n in range(1,301) if abs(mobius(divisor_sigma(n, 1))) == 1] #* or *#

%o [n for n in range(1,301) if core(divisor_sigma(n,1)) == divisor_sigma(n,1)] # _Indranil Ghosh_, Mar 19 2017

%Y Cf. A000203 (sigma), A002110, A005117, A008683 (mu), A065299.

%K nonn

%O 1,2

%A _Labos Elemer_, Oct 29 2001