A341940 Numbers m such that phi(m)*tau(m) is a square but phi(m)/tau(m) is not the square of an integer.

54, 1026, 1280, 2187, 2304, 3840, 4352, 6750, 8802, 9072, 9900, 12500, 13056, 13718, 17496, 18700, 21870, 25856, 36900, 37500, 41154, 41553, 47682, 50432, 56100, 57078, 65792, 69700, 77568, 78786, 79200, 84240, 100000, 102656, 103586, 111100, 117666, 125712

Offset: 1

Author

Bernard Schott, Feb 24 2021

Keywords

• nonn

Comments

If phi(m)/tau(m) is a square of an integer (m is in A341939) then phi(m)*tau(m) is also a square (m is in A341938), but the converse is false. This sequence consists of these counterexamples (see the Examples section).

Examples

			phi(54) = 18, tau(54) = 8, phi(54)*tau(54) = 18*8 = 144 = 12^2 but phi(54)/tau(54) = 9/4 = (3/2)^2 is not the square of an integer, hence 54 is a term.
phi(1026) = 324, tau(1026) = 16, phi(1026)*tau(1026) = 324*16 = 5184 = 72^2 but phi(1026)/tau(1026) = 324/16 = 81/4 = (9/2)^2 is not the square of an integer, hence 1026 is another term.
		

Crossrefs

Similar for: A327624 (phi(n) and sigma(n)), A327831 (sigma(n) and tau(n)).

Equals A341938 \ A341939.

Cf. A000005 (phi), A000010 (tau).

Programs

Maple

with(numtheory): filter:= r -> phi(r)/tau(r) <> floor(phi(r)/tau(r)) and issqr(phi(r)*tau(r)) : select(filter, [$1..50000]);

Mathematica

Select[Range[10^5], IntegerQ /@ Sqrt[{(e = EulerPhi[#])*(d = DivisorSigma[0, #]), e/d}] == {True, False} &] (* Amiram Eldar, Feb 24 2021 *)

PARI

isok(m) = my(x=eulerphi(m), y = numdiv(m)); issquare(x*y) && (denominator(x/y) != 1); \\ Michel Marcus, Feb 24 2021