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A341940 Numbers m such that phi(m)*tau(m) is a square but phi(m)/tau(m) is not the square of an integer. 2

%I

%S 54,1026,1280,2187,2304,3840,4352,6750,8802,9072,9900,12500,13056,

%T 13718,17496,18700,21870,25856,36900,37500,41154,41553,47682,50432,

%U 56100,57078,65792,69700,77568,78786,79200,84240,100000,102656,103586,111100,117666,125712

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

%C 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).

%e 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.

%e 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.

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

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

%o (PARI) isok(m) = my(x=eulerphi(m), y = numdiv(m)); issquare(x*y) && (denominator(x/y) != 1); \\ _Michel Marcus_, Feb 24 2021

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

%Y Equals A341938 \ A341939.

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

%K nonn

%O 1,1

%A _Bernard Schott_, Feb 24 2021

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Last modified June 23 18:45 EDT 2021. Contains 345402 sequences. (Running on oeis4.)