A327624 Numbers m such that sigma(m)*phi(m) is a square but sigma(m)/phi(m) is not an integer.

51, 170, 194, 364, 405, 477, 595, 679, 760, 780, 1023, 1455, 1463, 1496, 1512, 1524, 1674, 1715, 1731, 1796, 1804, 2058, 2080, 2651, 2754, 2945, 3192, 3410, 3534, 3567, 4381, 4420, 5044, 5130, 5670, 5770, 5784, 5797, 5822, 5859, 7600, 8245

Offset: 1

Author

Bernard Schott, Sep 19 2019

Keywords

• nonn

Comments

If sigma(m)/phi(m) is a square (m is in A293391) then sigma(m)*phi(m) is also a square (m is in A011257), but the converse is false (see 51 in the Example section). This sequence consists of these counterexamples.

Examples

			phi(51) = 32 and sigma(51) = 72, phi(51) * sigma(51) = 32 * 72 = 2304 = 48^2, but sigma(51)/phi(51) = 72/32 = 9/4 is not an integer.
		

Crossrefs

Equals A293391 \ A011257.

Cf. A020492 (sigma(m)/phi(m) is an integer).

Cf. A000010 (phi), A000203 (sigma).

Programs

Magma

[k:k in [1..9000]| not IsIntegral(SumOfDivisors(k)/EulerPhi(k)) and IsSquare(EulerPhi(k)*SumOfDivisors(k)) ]; // Marius A. Burtea, Sep 19 2019

Maple

filter:= v -> sigma(v)/phi(v) <> floor(sigma(v)/phi(v)) and issqr(sigma(v)*phi(v)) : select(filter, [$1..50000]);

Mathematica

sQ[n_] := IntegerQ @ Sqrt[n]; aQ[n_] := sQ[(p = EulerPhi[n]) * (s = DivisorSigma[1, n])] && !sQ[s/p]; Select[Range[10^4], aQ] (* Amiram Eldar, Sep 19 2019 *)

PARI

isok(m) = my(s=sigma(m), e=eulerphi(m)); issquare(s*e) && (s%e); \\ Michel Marcus, Sep 21 2019