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

232, 2152, 3240, 3560, 3944, 6516, 17908, 22504, 23716, 26172, 32360, 34344, 36584, 37736, 43300, 45612, 48204, 55080, 55912, 60520, 61480, 69352, 73084, 78184, 79056, 79300, 96552, 104168, 105832, 106088, 125356, 130432, 133864, 140040, 149992, 163764, 168424, 172840, 176360, 183204

Offset: 1

Author

Bernard Schott, Oct 14 2019

Keywords

• nonn

Comments

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.

It seems that all terms are even. - Marius A. Burtea, Oct 15 2019

Examples

			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.
		

Crossrefs

Equals A144695 \ A327830.

Similar to A327624 with sigma(m) and phi(m).

Cf. A003601 (sigma(m)/tau(m) is an integer), A023883 (sigma(m)/tau(m) is an integer and m is nonprime).

Cf. A000005 (tau), A000203 (sigma).

Programs

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

Maple

filter:= u -> sigma(u)/tau(u) <> floor(sigma(u)/tau(u)) and issqr(sigma(u)*tau(u)) : select(filter, [$1..100000]);

Mathematica

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 *)

PARI

isok(m) = my(s=sigma(m), t=numdiv(m)); issquare(s*t) && (s % t); \\ Michel Marcus, Oct 15 2019