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A327624
Numbers m such that sigma(m)*phi(m) is a square but sigma(m)/phi(m) is not an integer.
3
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
1,1
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.
EXAMPLE
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.
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 *)
PROG
(Magma) [k:k in [1..9000]| not IsIntegral(SumOfDivisors(k)/EulerPhi(k)) and IsSquare(EulerPhi(k)*SumOfDivisors(k)) ]; // Marius A. Burtea, Sep 19 2019
(PARI) isok(m) = my(s=sigma(m), e=eulerphi(m)); issquare(s*e) && (s%e); \\ Michel Marcus, Sep 21 2019
CROSSREFS
Equals A293391 \ A011257.
Cf. A020492 (sigma(m)/phi(m) is an integer).
Cf. A000010 (phi), A000203 (sigma).
Sequence in context: A257352 A353599 A008883 * A069762 A245829 A369922
KEYWORD
nonn
AUTHOR
Bernard Schott, Sep 19 2019
STATUS
approved