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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..42.

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: A044764 A257352 A008883 * A069762 A245829 A031431

Adjacent sequences:  A327621 A327622 A327623 * A327625 A327626 A327627

KEYWORD

nonn

AUTHOR

Bernard Schott, Sep 19 2019

STATUS

approved

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Last modified June 23 08:51 EDT 2021. Contains 345395 sequences. (Running on oeis4.)