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A327831 Numbers m such that sigma(m)*tau(m) is a square but sigma(m)/tau(m) is not an integer.

%I A327831 #18 Sep 08 2022 08:46:24
%S A327831 232,2152,3240,3560,3944,6516,17908,22504,23716,26172,32360,34344,
%T A327831 36584,37736,43300,45612,48204,55080,55912,60520,61480,69352,73084,
%U A327831 78184,79056,79300,96552,104168,105832,106088,125356,130432,133864,140040,149992,163764,168424,172840,176360,183204
%N A327831 Numbers m such that sigma(m)*tau(m) is a square but sigma(m)/tau(m) is not an integer.
%C A327831 If sigma(m)/tau(m) is a square (m is in A144695) then sigma(m)*tau(m) is also a square (m is in A327830), but the converse is false (see 232 in the Example section). This sequence consists of these counterexamples.
%C A327831 It seems that all terms are even. - _Marius A. Burtea_, Oct 15 2019
%e A327831 sigma(232) = 450 and tau(232) = 8, so sigma(232)*tau(232) = 450*8 = 3600 = 60^2 and sigma(232)/tau(232) = 450/8 = 225/4 is not an integer, hence 232 is a term.
%p A327831 filter:= u -> sigma(u)/tau(u) <> floor(sigma(u)/tau(u)) and issqr(sigma(u)*tau(u)) : select(filter, [$1..100000]);
%t A327831 sQ[n_] := IntegerQ@Sqrt[n]; aQ[n_] := sQ[(d = DivisorSigma[0, n]) * (s = DivisorSigma[1, n])] && !sQ[s/d]; Select[Range[2*10^5], aQ] (* _Amiram Eldar_, Oct 15 2019 *)
%o A327831 (PARI) isok(m) = my(s=sigma(m), t=numdiv(m)); issquare(s*t) && (s % t); \\ _Michel Marcus_, Oct 15 2019
%o A327831 (Magma) [k:k in [1..200000] | not IsIntegral(a/b) and IsSquare(a*b) where a is DivisorSigma(1,k) where b is #Divisors(k)]; // _Marius A. Burtea_, Oct 15 2019
%Y A327831 Equals A144695 \ A327830.
%Y A327831 Similar to A327624 with sigma(m) and phi(m).
%Y A327831 Cf. A003601 (sigma(m)/tau(m) is an integer), A023883 (sigma(m)/tau(m) is an integer and m is nonprime).
%Y A327831 Cf. A000005 (tau), A000203 (sigma).
%K A327831 nonn
%O A327831 1,1
%A A327831 _Bernard Schott_, Oct 14 2019