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A065299 Numbers k such that sigma(k)*phi(k) is squarefree.

%I A065299 #21 Feb 06 2018 03:03:49
%S A065299 1,2,4,9,121,242,529,1058,2209,3481,4418,5041,6889,6962,10082,11449,
%T A065299 13778,17161,22898,27889,32041,34322,51529,55778,57121,64082,96721,
%U A065299 103058,114242,120409,128881,146689,175561,185761,193442,196249,218089
%N A065299 Numbers k such that sigma(k)*phi(k) is squarefree.
%H A065299 Harry J. Smith and Charles R Greathouse IV, <a href="/A065299/b065299.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Smith)
%F A065299 Solutions to abs(A008683(A000203(x)*A000010(x))) = 1.
%e A065299 All solutions are either squares or twice squares. Proper subset of A055008 or A028982. Several squares (of primes) and 2*squares are not here. E.g., 242 is here because phi(242) = 110, sigma(242) = 399, 2*5*11*3*7*19 is squarefree; 18 is not here, since 2*3*3*13 is not squarefree.
%t A065299 a[x_] := Abs[MoebiusMu[DivisorSigma[1, x]*EulerPhi[x]]] Do[s=as[n]; If[Equal[s, 1], Print[{n, Sqrt[n]}]], {n, 1, 1000000}]
%t A065299 Select[Range[250000],SquareFreeQ[DivisorSigma[1,#]*EulerPhi[#]]&] (* _Harvey P. Dale_, Jul 15 2015 *)
%o A065299 (PARI) n=0; for (m = 1, 10^9, s=abs(moebius(sigma(m)*eulerphi(m))); if (s==1, write("b065299.txt", n++, " ", m); if (n==500, return))) \\ _Harry J. Smith_, Oct 15 2009
%o A065299 (PARI) is(f)=my(n=#f~, v=List()); for(i=1,n, if(f[i,1]>2, listput(v,f[i,1]-1)); if(f[i,2]>2, return(0), f[i,2]>1, listput(v,f[i,1])); listput(v, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))); for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, if(!issquarefree(v[i]), return(0))); 1
%o A065299 sq(f)=f[,2]*=2; f
%o A065299 double(f)=if(#f~ && f[1,1]==2, f[1,2]++, f=concat([2,1],f)); f
%o A065299 list(lim)=my(v=List()); forsquarefree(n=1,sqrtint(lim\1), if(is(sq(n[2])), listput(v,n[1]^2))); forsquarefree(n=1,sqrtint(lim\2), if(is(double(sq(n[2]))), listput(v,2*n[1]^2))); Set(v) \\ _Charles R Greathouse IV_, Feb 05 2018
%Y A065299 A062354(m) is squarefree.
%Y A065299 Cf. A000010, A000203, A008683, A055008, A028982, A049195-A049199, A049149.
%K A065299 nonn
%O A065299 1,2
%A A065299 _Labos Elemer_, Oct 29 2001