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

1, 2, 4, 9, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 17161, 22898, 27889, 32041, 34322, 51529, 55778, 57121, 64082, 96721, 103058, 114242, 120409, 128881, 146689, 175561, 185761, 193442, 196249, 218089

Offset

1,2

Links

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..500 from Smith)

Formula

Solutions to abs(A008683(A000203(x)*A000010(x))) = 1.

Example

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.

Mathematica

a[x_] := Abs[MoebiusMu[DivisorSigma[1, x]*EulerPhi[x]]] Do[s=as[n]; If[Equal[s, 1], Print[{n, Sqrt[n]}]], {n, 1, 1000000}]
Select[Range[250000], SquareFreeQ[DivisorSigma[1, #]*EulerPhi[#]]&] (* Harvey P. Dale, Jul 15 2015 *)

Prog

(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
(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
sq(f)=f[, 2]*=2; f
double(f)=if(#f~ && f[1, 1]==2, f[1, 2]++, f=concat([2, 1], f)); f
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

Crossrefs

A062354(m) is squarefree.

Cf. A000010, A000203, A008683, A055008, A028982, A049195-A049199, A049149.

Sequence in context: A271553 A067603 A269739 * A292114 A128942 A364637

Adjacent sequences: A065296 A065297 A065298 * A065300 A065301 A065302

Keyword

nonn

Author

Labos Elemer, Oct 29 2001

Status

approved