

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



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
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OFFSET

1,1


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).


LINKS

Table of n, a(n) for n=1..38.


EXAMPLE

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.


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 *)


PROG

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


CROSSREFS

Similar for: A327624 (phi(n) and sigma(n)), A327831 (sigma(n) and tau(n)).
Equals A341938 \ A341939.
Cf. A000005 (phi), A000010 (tau).
Sequence in context: A298718 A245832 A121625 * A190399 A071803 A215836
Adjacent sequences: A341937 A341938 A341939 * A341941 A341942 A341943


KEYWORD

nonn


AUTHOR

Bernard Schott, Feb 24 2021


STATUS

approved



